Algebraic numbers and harmonic analysis

6.3. Test functions On a 1.c.a. group G, the test functions are the complex valued continuous functioris n which belong both to L1(G) and to A(G); by the inversion formula ([15], p 22, th. 1.5.1) the Fourier transform 4 of n also has these two properties. In [15] it is proved that we can construct partitions of unity by means of these test functions. The vector space of test-functions will be denoted by d(G).

If y belongs to L m ( r )the Fourier transform S = @ of y is a distribution S defined by (S, a) = Jr y(y) &) dy for each test function ol in d(G). Since this integral is convergent for all & in L1(r), it is sufficient to assume that ol E A(G). Such an Sis called a pseudo-measure. We put 11 Sll., = llyll and, with this norm, the pseudo-measures form a Banach space denoted by PM (G). In the preceding terminology, the spectrum of y is the support of the pseudo-measure @. Note that each p in M(G) is a pseudo-measure; in this case y(y) = JG xy(x) dp (x) and p is the Fourier transform of the complex valued bounded continuous function y. 6.5. Compactly supportedpseudo-measures For a compactly supported pseudo-measure S, we can define a version of the Fourier transform $of S, which is a complex valued bounded continuous function on if $ = y E L m ( r ) , let n be an element of d(G), which is equal to 1 on a neighborhocd of the support of S and write S = Sol; hence g, = 3 = * = y is the required continuous bounded

r:

UNIQUENESS OF REPRESENTATION

UNIQUENESS OF REPRESENTATION

function, where the equality is an almost everywhere equality and y is the convolution of an element of L1(r) and an element of L w ( r )and is therefore continuous.

The dual group of Q, is Q , itself; each continuous character X : Q , -+ T is of the form ~ ( x = ) exp 2nixy for a suitable y in Q,. Let 8 be an element of Q , such that 181, > 2. Let E(8) be the compact set of all convergent sums ~ , 8 - ~ , = 0 or 1.

6.6. The scalar product of a compactly supported pseudo-measure S a n d a function g in B(G) is defined using the same function a : ag belongs to A(G) and ( S , g ) = ( S , a g ) ; this definition is independent of the particular a chosen.

7.2. THEOREM VI: The compact set E(0) is a set of uniqueness ifand only if 8 is a Pisot-Chabauty number. The proof of the necessity of the condition is very similar to the proof given for the real case. We can construct as in Section 2.2 a probability measure ,u carried by E(8), whose Fourier transform is

100

C:

*

6.7. Homomorphisms of groups KY) Let GI and G, be two 1.c.a. groups and h : GI + G, a continuous homomorphism. For each f in A(G,), f 0 h = g belongs to B(Gl) and IlgllBcc,) < I l f llacc,, ([151, 4.2.1, P. 79). Hence for each compactly supported pseudomeasure S on G I , we can define a pseudo-measure h(S) on G, by the rule
6.8. Sets of multiplicity A compact subset E of G is said to be a set of multiplicity if there exists a non zero pseudo-measure S carried by E whose Fourier transform vanishes at infinity. An equivalent condition is that there exists a continuous complex valued function f : I'+ C vanishing at infinity whose spectrum lies in E.

7. The p-adic case 7.1. Let Q , be the p-adic field, A, be the subring of Q , consisting of all mpn,m E Z , n E Z , m a n d p relatively prime. Thep-adic norm is defined on A , by Impn], = p-" and for this norm A, is dense in Q,. But A, is also a subring of R and the norm (mpnl, is induced on A, by the Archimedean norm of R. Let Z , be the subring of all x in Q , such that 1x1, < 1 : Z , is the completion of Z with respect to the p-adic norm. Each x in Q , can be written uniquely in the form x = y + r where y E Z,, r E A, and 0 < r < 1. We put exp 2nix = exp 2nir.

101

=

n1+ 1

exp 2iO- ky 2

To study the behaviour of F at infinity, we divide Q, into the cells 181; < lyl, < 181:'~ which leads one to put y = Oku and k- 1

1

+ exp 2i@u

0

We get, if y = Okuand 1 < \ulp < IOl,, I$(y)l < Pk(u).If 0 is not a PisotChabauty number, Pk(u)tends uniformly to 0 on each compact subset of Q, not containing 0 by Pisot-Chabauty's theorem. Hence ok= sup 1 4 U l ~ q P

IPk(u)/ tends to 0 as k tends to infinity and, for 101: @(Y)I < m k .

< Iyl, < lOli'

,

7.3. E(8) is a set of uniqueness when 8 is a Pisot-Chabauty number

xkaO

Let A be the set of all p-adic number ekek,where ek = 0 or 1 and where the sum is finite. The proof is implicitly contained in the statement that A is a part of a model. Sketch of the proof: Let S be a pseudofunction carried by E such that g(0) = 1. Let Sk be defined by Sk(x) = F k S ( 8 - k ~ )Skis ; then carried by ekE c A + E. (a) there is a compact group G and a continuous injective homomorphism h : Q , -+ G with the following properties: (1) h(Q,) is dense in G ; (2) there is a compact subset K of G of measure 0 such that h (A E ) c K;

+

102

UNIQUENESS O F REPRESENTATION

UNIQUENESS OF REPRESENTATION

(b) Putting T k = h(Sk),we get a bounded sequence of pseudo-measures carried by Kand converging in the weak star topology of pseudo-measures to the limit dx, where dx is the Haar measure of G. This is impossible. Details of the proof: Part (b) of the proof is an easy modification of the corresponding argument used in the real case. We have to check (a). Let n be the degree of 8, P the field Q(8) and 8 the product group Q p x R n ;w e w r i t e ( x , y ) ~ @ i f x ~ Q , a n d y ~Let R ~l ,. a 2 , ..., a , b e a + a,03-~ dualbasisofthebasis(1, 0, ..., 8"-l)definedby 1 a20j + = 0 for 2 < j Q n, I?,, ..., 8, b e i ~ gthe coajugates of 8 over Q . Then a,, ..., a, lie in 9 c Q , and, provided c > 0 is large enough, A is c o n t a i ~ e din the set M of all elements A of 9 of the form A = t , + t28 + t,On-l for all t , , . .., t, in A, satisfying the inequalities: It2 - a2tllp < c, ..., It, - antlJp< c, It,l, < c, ..., Itnl, < c (C1napterII, Section 10). Let D be the discrete and relatively dense subgroup of Q , x Rn consisting of all ( x , y) of the form x = t , + t28 ... tnBn-', y = ( t , , t 2 , ..., tn)for t , , ..., tn in A, such that It2 - a2t,lp < c, ..., It, - a,t,l, Q c. Let G be the quotient group @ID, li' the canonical projection of 8 onto G and h the restriction of L7 to Q,. Then the image of Q , under h is a dens: subgroup of G. Definition of K: Let B c Rn be the set of all y = ( y , , . .., y,) in Rn such that lyllm < c, ..., lynlm < C. Then K = n ( E x B). Since D is is measure-decreasing. Hence, if I I denotes the cordiscrete in @, responding Haar measures with suitable normalizations, JKI < IE x BI = JEl 1B1 = 0 since E has zero measure in Q,. h ( A + E ) is contained in K. Since h is the restriction of to Q, , it suffices to show that A + E c E x B + D. But for each A in A, we can find a y in B such that (A, 0) + (0, y) E D or, since -B = B, a y in B such that (A, 0) = (0, y) (mod D). If t E E, then ( A + t, 0 ) = (t, y) (mod D) and the proof is complete.

+

- a -

+

+

+

n

8. Generalizations As in the real case, the proof can be extended to symmetric sets constructed with variable dissection ratios whose inverses are 'uniform' Pisot numbers of a given algebraic number field contained in Q,.

THEOREM VII: Let P be an extension of the jield Q of finite degree n contained in Q , and let z , , . .., z, be the n Q-isomorphismsfrom 9 into C . Let 0 < a < 1 , and let (co,),, be a sequence of Pisot-Chabauty numbers

,

103

of degree n in R satisfying the condition of uniformity: Izj(cok)l, < a, for all 1 < j Q n and k > 1, and such that llcokllP> 2 for all k 2 1. Then the symmetric set E constructed with the sequence (ok-') of dissection ratios is a set of uniqueness. The definition of E is the same as that given in the real case : E is the set of all sums Cy s k t k ,E~ = 0 or 1, tk = 5, t k - 1 (1 - t k ) . 0

.

.

9. Other results on the problem of uniqueness In Chapter VII ultrathin symmetric sets will be studied: when the dissection ratios tend to 0 sufficiently rapidly, no arithmetical properties are needed to get sets of uniqueness. The exact condition is Cy 5; < + + and it is the best possible in the sense that the exponent 2 cannot be replaced by a larger one.

10. Notes

A remarkable survey on the problem of uniqueness and multiplicity is given in [2], Ch. XIV. A first version of Theorem IV may be found in [lo], where there is a requirement of periodicity on the sequence of dissection ratios. For other results, the reader is referred to [18], 191, [58] and [92].

105

PROBLEMS ON A-PERIODIC TRIGONOMETRIC SUMS

CHAPTER IV

natural to expect f(to, t) to be a good approximation of f(t) in a suitable neighborhood of to. Surprisingly, two extra conditions are needed to obtain a good result: an a priori estimate of the behaviour off at infinity (I f(t)l < C exp ItuI, 0 < a < 1, is sufficient) and a remarkable arithmetical structure for A (it is sufficient to assume that A is a coherent set of frequencies).

PROBLEMS O N A-PERIODIC T R I G O N O M E T R I C SUMS

1. Elementary properties of almost-periodic functions Almost periodicity is a useful and natural generalization of periodicity: a bounded continuous f : R + C is almost periodic if for each positive E , there exists a T(E) > 0 such that each interval of length T(E) contains an &-almostperiod z off, i.e. sup I f(t z) - f(t)l < E sup If 1.

+

R

R

The theory shows that T(E) depends on f in a very intricate way: to make any estimate of T(E) one needs to have knowledge off right up to the end of time and T(E) cannot be used to predict what f will be. The general theory only gives a posteriori estimates of &-almostperiods. This is quite restricting and our first problem is to get a priori estimates of &-almostperiods for reasonable spaces V of almost periodic functions. By reasonable spaces V, we mean spaces that are translation invariant and linear, and in this case the answer to our first problem is given by the following rule: the frequencies of any f in V belong to a given harmonious set. A second problem is to give a priori estimates of the size of almost periodic functions f belonging to such reasonable spaces, where the size off is taken to be sup (f1. We are led to the following problem: letA be a R

set of real numbers; do a compact set K and a constant C exist such that for any trigonometric sum whose frequencies belong to A, sup If ( < C R

sup If I? If they do exist, A is said to be a coherent set of frequencies. K

A third problem is to study the effect of slow perturbations on trigonometric sums whose frequencies belong to a given set A of frequencies. Perturbations are said to be slow when constant coefficients are replaced by continuous functions whose spectra are contained in a small neighborhood of 0: in the real case, we get small frequencies and great periods as in the secular perturbations of astronomy. Iff(t) = al(t) exp 2niAt is such a slowly-perturbed trigonometric sum and if f(to, t) = al(to) exp 2niAt is the corresponding family of pure trigonometric sums, it is

Elen

Enen

We are mainly interested in spaces of trigonometric sums defined on a 1.c.a. group G; a trigonometric sum P is a sum ayXyof continuous characters with complex coefficients indexed by the dual group such that the complex numbers a, are equal to 0 for all y in r with the exception of at most a finite number of y; the set of all y E ~ S U that C ~ a, f 0 is called the spectrum of P or the set of frequencies of P. In the real case, P(t) = a,, exp 2nivt and the real numbers v for which a, # 0 are the frequencies of P. A great number of properties of such sums hold for uniform limits on G of trigonometric sums. We have just defined the space of almost periodic functions on G. We recall without proofs the elementary properties of complex valued almost periodic functions on G. The reader is referred to [ l l ] for the proofs. Let Vb(G) be the Banach space of all complex-valued bounded continuous functions on G. The norm off E Vb(G) is 11 f 11 = sup (f(x)l and XEG is also written sup If 1.

Eyer

r

EYER

G

For each f E Vb(G) and each s E G, the translated functionf, is defined by f,(x) = f(x - s) for all x in G; for each positive E , the set M (E,f ) of oalmost periods off is defined by s E M ( E , f ) if Ilf, -f 11, < f 11 ,This is a slight modification of the usual definition: the normalization by (1 f 11, is fundamental in the following.

.

DEFINITION 1 : A function f E Vb(G)is said to be almost-periodic ifone of the following two equivalent conditions is satisfied. (I) for each positive E, the set M ( E , f ) of &-almostperiods off is relatively dense; (2) for each positive E, there exist a finite subset F o f r and a trigonometric sum P(x) = ag,(x) such that Il f - Pll, < e. I f f is almost periodic, some precise information can be given about

I,,,

PROBLEMS O N A-PERIODIC TRIGONOMETRIC SUMS

PROBLEMS ON A-PERIODIC TRIGONOMETRIC SUMS

the sets F used in (2). The closed linear subspace of Vb(G)spanned by all thef,, s E G, is denoted by E(f).

2.2. In an attempt to generalize this situation, we are led to the following definition :

106

DEFINITION 2 : The spectrum, denoted by Sp ( f ) , of an almost periodic function f on G is the set of all y in the dual group of G such that the corresponding character X, belongs to E ( f ) . The theorem of spectral synthesis for almost periodic f~~nctions is an improvement of property 2 in Definition 1.

r

THEOREM I: Let f be a complex-valued almost periodic function deJned on an 1.c.a. group G a n d A be the spectrum o f f . Then there exists a sequence ( P k ) k 2 1of trigonometric sums whose frequencies belong to A such that At this point, it seems that the theory of almost periodic functions is going to be quite easy-going: we can define the Fourier series of almostperiodic functions, the Parseval relation is valid, and so on. How do new problems arise?

2. Classes of almost-periodic functions 2.1. We come back to the real line. The space, denoted by AP ( R ) , of all complex-valued almost-periodic functions on the real line is a huge space containing all complex-valued continuous periodic functions; indeed, AP ( R ) is not separable. The properties of an individual almostperiodic function on the line are very similar to the properties of an individual periodic function, but the common properties of all complexvalued continuous 2n-periodic functions are very different from the common properties of all almost periodic functions We shall list these properties :

DEF~NITION 3 : Let %? be a linear subspace of AP (G). We say that V is a class of almost-periodic functions i f V has the following three properties: (a) %? is closed in AP (G); (b) %? is translation-invariant; (c) for all positive F , there exists a relatively dense set M ( F )in G such hat for each s in M(E)and each f in 9, [ I f , f [ I , < ~ l l/ Ifm . Example : the space of all complex valued continuous 2n-periodic functions on the line is a class of almost-periodic functiors.

r

THEOREM 11: Let A be a harmonious subset of the dual group of a 1.c.a. group G. Let V(A)be the linear subspace of all almost-periodic functions on G whose spectra lie in A. Then %(A) is a class of almost periodic functions and each class of almost-periodic functions is a %?(A) for a suitable harmonious set A. It will be shown in the interlude that (4.4) holds for V ( A )if A is relatively decse in and if K is a suitable compact neighbourhood of 0.

r

2.3. Proof of theJrst half of Theorem I1 LEMMA1 : Let I be a real number in the open interval 10, n[ and a : R + [- 1, 11the continuous 41-periodicfunction dejinedby a(t) = a(- t )for all real t and a(t) = sin ( I - t ) / 2 i f 0 < t < 21. The Fourier expansion of ,a(t) is c, cos (k + t )~ 1 - It , where c, > 0 and c, = sin 112. Proof: An easy computation gives

Z:

Each t E 2nZ is a period for each 2mperiodic function. The 2n-periodic functions are given by their values on the compact subset [O,2n] of R. The space of all continuous 2n-periodic functions is closed in V b ( R ) and is translation invariant. If K is a compact subset of [O,2n], different from [0, 2n], each complex-valued continuous function on K can be extended to a 2n-periodic function.

107

2:

Ck =

21 cos 112 > o. (2k + 1)' n2 - 1'

2:

Hence c, < + oo, the function is the sum of its Fourier series and c, = a(0) = sin 112.

Z,"

LEMMA 2 : Let p = 2q + 1 be an odd integer and I = n / p There exists a sequence (dn)-, ,,,+, of complex numbers such that (a) Idn! = leil - 11 = 2 sin 112; (b) .if - I < x < I, eix - 1 = dn einx.

z?:

x?:

109

PROBLEMS O N A-PERIODIC TRIGONOMETRIC SUMS

PROBLEMS O N A-PERIODIC TRIGONOMETRIC SUMS

Proof: We write eix - 1 = eix" (eiX1' - e-ixlz) = 2i elXi2sin x/2. Put t = 1 - x and apply Lemma 1. The hypothesis that p is odd and 1 = n / p ensures that each e-Iti2 cos (k + +) nl-l t can be written eint, for some n E Z. It can be shown that the hypothesis 1 = n/p, p odd is necessary for (a) and (b) to hold.

Proof: Let A be the union of' all Sp ( f )for allf in V . Definition 2 shows that Xn belongs to V for each ilin A, and Theorem I shows that each f in A P (G) whose spectra lie in A belongs to V If V is a class of almost-periodic functions, we get Ix,(x) - I / < E for each x in M(E)and each il in A; since M ( E )is relatively dense, A is harmonious

108

.

3: For each real number 6 in [0, 11, there exists a sequence LEMMA (d,) -, ,,+, of complex numbers such that (a) 1d.l < 26; (b) lzI = 1 and lz - 11 < 6 imply z - 1 = dnzn. Proof: I is defined by 1 E [O,76/31 and sin 112 = 612, a n d p is the greatest odd integer less than rill. Then p > 3 and Lemma 2 yields the sequence + m Idn/ = 2 sinnl2p. more over,^ < n/l < p + 2 (dn)-mcn<+, such that < 2p implies sin n/2p < sin I < 2 sin 112 = 6.

1;:

CZg

r.

LEMMA 4: Let G be a 1.c.a. group, T the dual group a n d A a subset of Each element s of G such that sup (~,()3) - 11 = 6 < 1 is a 26-almost A

period of each almost-periodic function f whose spectrum lies in A. Since the subset of AP (G) defined by I l f , - f 1 1 , < 2611f ll ,is closed, Theorem I shows that we can restrict our attention to trigonometric sums whose frequencies belong to A. We define the sequence (dn)nEZ by Lemma 3 and a discrete measure o on G by o = dnen,, where E, is the unit point-mass at x. Then for each il in A, JG d~ = Cf_gdnxr (ns) = d. (xA(s))" = x&) - 1. Assume that f is a finite trigonometric sum cIxr;then f(s) - f(0) = Enen CI (xL@)- 1) = C I JG Xr do = J G do ~ implies I f ( 8 ) - f (011 < 26jlf 11,. To get the inequality I f ( s + x ) -f(x)l < 2611f 11, foreachxin G, we apply the preceding proof to the function g defined on G by t -+ f ( t +x). We now return to Theorem 11. I f A is harmonious, there is, for each positive 6, a relatively dense subset N(6) of G such that sup (x,(s) - 11 < 6

x+z

1 : :

ZI~A

CrEA

A

for each s E N(6). Lemma 4 gives Ilf, - f of Theorem I1 is proved [91].

1 1 , < 26 11 f 11 ,and the first part

2.4. The proof of the second part depends on the following lemma: 5: Let V c AP (G) be a closed linear subspace which is transLEMMA lation-invariant. Then there exists a subset A of the dualgrouprsuch that V is the set of all f in AP (G) whose spectra lie in A.

3. Dilations and classes of almost-periodic functions on the line Let 8 > 0 be a positive real number. The dilation with ratio O is defined on R by x + Ox. Iff is any complex-valued function on R, Tof will be the complex-valued function g defined by g(x) = f (Ox), rather than the expected f (0-'x). The product of two dilations is clearly another dilation and if V is a set of complex-valued functions on the line, we shall study the set S of all 8 > 0 such that To(%)c V . THEOREM 111: Let V be a vector space of almost-periodic functions on R. Assume that (a) V is translation-invariant; (b) for each positive E there is a relatively dense subset M ( E )of R such that each s E M ( E )is an E-almostperiod for all f in V ; (c) there is a function f in V which is not a constant. Let (To)oES be the semigroup of all dilations such that To(%)c V . Then there exists a real algebraic numberjeld K, of degree n over Q say, such that S is contained in (1) n 2, where is the set of all Pisot or Salem numbers of degree n lying in K. Before proving this theorem, a remark should be made. It may happen that the only 8 for which To(% ' ?) c V is O = 1. All the other such O are Pisot or Salem numbers. Proof: Let C be the closure of V with respect to the uniform norm. Then has the properties (a) and (b) and, using Theorem 11of Chapter IV, %?is the space of all almost-periodic functions f on the line whose spectra lie in an harmonious set A of real numbers. If O E S and O # 1 , we apply the inclusion To(@)c C to each element exp 2niAx, ilE A, of 8;we get 8A c A. Since there is a function f in V which is not a constant, there is a 2 in A which is not zero. Hence Sil c A is harmonious and so is S. We apply Theorem VI of Chapter I1 (Section 8) to S.

110

PROBLEMS ON A-PERIODIC TRIGONOMETRIC SUMS

PROBLEMS ON A-PERIODIC TRIGONOMETRIC SUMS

4. Classes of almost-periodic functions and a priori estimates on the size

r

We shall see that each harmonious subset A of the dual group of G is a coherent set of frequencies; the converse, however, is not true. PROPOSITION 1 : Let V be a class of almost periodic functions, M(E) a relatively dense set of s-almost periods for all f in V, and K(E) a compact subset of G such that K(E) + M(E) = G. For each f in V, sup If ( 2 (1 - E ) K.

Ilf ll m .

This means that, within a relative error not exceeding E, the functions in V are 'known' on the whole group as soon as they are known on K,, just as the 2n-periodic functions on the line are fully determined by their values on [O, 2n]. The proof of Proposition 1 is very simple. Each y in G can be written y = s + x, where s E M(s), x E Ke. Since each s E M(s) is a n &-almost period for each f in V, we get I f(y) - f(x)l < film; hence (f(y)l < sup (f( + f 11 a r d taking the supremum over y E G we get ProposiK.

tion 1. Let G = R and let (tk)k,O be an increasing sequence of positive real numbers such that (tk/tk+1)2< CO. Let A be the set of all finite where E, = 0 or 1. We shall prove (Chapter VIII, Secsums tion 5, Theorem IV) that, for each positive E,there exists a compact subset Ke of R such that sup If 1 2 (1 - E) [] f /Im for each almost periodic

FkaO

XkaO

+

K ,

function f whose spectrum lies in A. However, the sequence (tk)kSOcan be chosen in such a way that A is not harmonious (Chapter VIII, Theorem XI). 5. Pisot numbers and coherent sets of frequencies: the real case THEOREM IV: Let 8 be a real number greater than one andA, the set of all finite sumsxkgO&,Ok, where E, = 0 or 1. Then the following two properties of 8 are equivalent: (a) 8 is a Pisot number; (b) A, is a coherent set of frequencies. Before proving Theorem IV, some remarks should be made : if 8 = 1, A, is the set N of all positive integers and A, is a coherent set of frequencies; if 8 = - 1, A, = Z and is a coherent set of frequencies. If 8 < - 1 the proof given below will show that A, is a coherent set of frequencies if

111

and only if - 8 is a Pisot number. If - 1 < 8 < 1, Proposition 3 below shows that A, cannot be a coherent set of frequencies. Now to the proof. (a) Assume 8 is a Pisot number: then A, is harmonious and hence A, is a coherent set of frequencies; (b) Now assume 8 is not a Pisot number. The following lemma will be needed. LEMMA 6: If 8 is not a Pisot number there exists a sequence (Pk)k,O of Jinite trigonometric sums whose frequencies belong to Ae such that

(2) on each compact subset K of R not containing 0, P, tends uniformly to 0 as k tends to infinity. Proof of the lemma: Let Pk(t) = exp nit (1 + + 8,) cos n t cos nOkt; Pk(t) is a finite trigonometric sum of the type an exp 2ni;lt. If 8 is not a Pisot number, IP,(t)l = lcos ntj lcos nOktl is a decreasing sequence tending to 0 for each t # 0: if we write IcosnOkt12= 1 - sin2 n02t, this fact is a consequence of Pisot's theorem. Dini's theorem shows that (Pk)kaOtends uniformly to 0 on each compact set of real numbers not containing 0.

xnen

5.1. Proof of Theorem I V Assume that there exist a compact set K of real numbers and a positive C such that, for each trigonometric sum whose frequencies belong to A,, sup IP( < C sup IPI. Let s be a real number such that s K does not R

+

K

contain 0 and set Qk(t) = Pk(S + t). Then Q, is a finite trigonometric sum whose frequencies belong to A, sup JQkl-+ 0 as k + + co and K

IIQkllm= 1. Hence A, is not a coherent set of frequencies.

5.2. Improvement on the preceding result Let 8 be a Pisot number, q a real number in 10, 1[ and T(q) the infimum of those positive T for which sup IP(t)l 2 (1 - q) ll Pll for each ItkT

trigonometric sum P whose frequencies belong to A,. Two problems can be stated: what is the behaviour of T(q) as 11 t e ~ d s to O? What is the behaviour of T(q) as 7 tends to I ?

112

PROBLEMS ON A-PERIODIC TRIGONOMETRIC SUMS

PROPOSITION 2: Let n be the degree of the Pisot number 8. There exist two positive constants a and b depending on 8, such that, for each q in 10, 1[, (-n+l)/z < , T(q) < bq(-nf1)/2. all This is the best possible result as Proposition 3 of Chapter I shows. This remarkable fact can be proved by the methods used in Chapter I, Section 4, and with the definition of models given in Chapter 11, Section 9. Observe that when q tends to 0, the quality of the prediction of the size of trigonometric sums whose frequencies belong to A increases at a corresponding 'cost', here the size of T(q), which is of the order of magnitude of v - ( " - ~ ) ' ~ . On the other hand when q tends to 1, the quality of the prediction decreases while the 'cost' of the prediction also decreases. What is the lower bound of this 'cost'? This is an open question. When q tends to 1, a natural conjecture is lim T(q) = DensA,. The density ofA, is defined V-+l

to be the uniform limit of T - l Card (A n [t, t + TI) as T tends to infinity. If 8 > 2 this density is 0, if 8 = 2 it is 1 and if 8 = (I + 4 5 1 2 (the golden number of Greek mathematicians) it is 3 + 3 &/lo. This conjecture will be provedin the case where Bis the quadratic Pisot number (1 + JT)/2 (Chapter V, Theorem 11).

113

Before proving Theorem IV, some remarks should be made :if 181, < 1, A, is an infinite subset of the unit ball of Q,. By Proposition 3 below, A, cannot be a coherent set of frequencies. We can therefore restrict our attention to those 8 in Q, such that 101, > 1. Now to the proof. (a) Assume 8 is a Pisot-Chabauty number: A, is harmonious and hence A, is a coherent set of frequencies; (b) Now assume 0 is not a Pisot-Chabauty number. The following lemma will be needed:

7.2. LEMMA 7: If 8 is not a Pisot-Chabauty number, there exists a sequence (P,),,, ofjinite trigonometric sums whosefrequencies belong to A, such that (1) sup lpkl = Pk(0) = 1 . QP

(2) on each compact subset K of Qp not containing 0, P, tends uniformly to 0 as k tends to infinity. Proof of the lemma: We recall that eachp-adic number can be written uniquely as a sum x = 5 r, where E E Z, and -3 < r < 3,r E A,, where A, is the subring of Q generated by p-'. We put r = p,(x) and the characters on Q, are of the form x -+ exp 2nipP(xA) for a suitable A in Q,. A finite trigonometric sum on Q, can be written a* exp 2nipp (AX). For each k 2 1, IetF, be the subset ofall sumsEk,-'~~8j, = 0 or 1 inA, andlet P, be the trigonometric sum2- '&,FK exp 2n@, (Ax). Then P,(x) can be written 2(1 exp 2nip, (Bjx)) and lPkl = Ilk-' ICOS n ~ (8jx)l. , Chabauty's theorem (Proposition 12, Section 12, Chapter 11) shows that if 8 is not a Pisot-Chaubauty number, P,(x) tends to zero as k tends to infinity for each x # 0. The proof is concluded as in the real case.

+

CAEap

6. A set of powers Let 8 be a real number greater than one and A the set of all powers Ok, k >/ 0. It will be proved that A is always a coherent set of frequencies (Chapter VI, Theorem 11). However, some problems remain open. Conjecture: Let 8 and A be as above. Then the following two properties are equivalent: (a) 8 is a Pisot or a Salem number; (b) for each positive q one can find a T(q) > 0 such that SUP Ip(t)l 2 (1 - 7) SUP Ip(t>l ItlG T ( q )

PROBLEMS ON A-PERIODIC TRIGONOMETRIC SUMS

'nk-'

+

R

for each trigonometric sum P whose frequencies belong to A.

8. The groups C and R2

7. Pisot numbers and coherent sets of frequencies: the p-adic case

Complex Pisot numbers can be defined in the following way: a complex number 0 is a complex Pisot number if there exists an irreducible element P(X) = X n + a,Xn-' + ... + an of Z[X] with the following properties : (a) P(0) = P@) = 0 and 8 $- 8; (b) the other n - 2 roots z of P(z) = 0 satisfy lzl < 1.

7.1. THEOREM V: Let 8 be up-adic number andA, the set of allfinite sums Zk,oskBk, where E , = 0 or 1. Then the following two properties of 8 are equivalent: (a) 8 is a Pisot-Chabauty number; (b) A is a coherent set of frequencies in Q,.

114

PROBLEMS ON A-PERIODIC TRIGONOMETRIC SUMS

It can be proved [43] that for a complex number 8, which is not real, the following two properties are equivalent: (a) 8 is a complex Pisot number; (b) 8 is neither purely real nor purely imaginary and the set A, of all finite sums ,, .skek,E~ = 0 or 1, is a coherent set of frequencies. Pisot pairs (a, 16) in R2 are defined in the following way: an element (a, p) in R2 is a Pisot pair if there is an irreducible element P(X) + a, of Z[X] with the following properties: = X" + a,Xn-l + (a) P(a) = P(P) = 0; (b) the other n - 2 roots z of P(z) = 0 satisfy lzl < 1. It can be proved [43] that for an element (a, p) of R2, the following two properties are equivalent : (a) (a,p) is a Pisot pair or la1 and 1161 are two Pisot numbers; (b) the set A& of all finite sums ( z k a OE ~ O L ~ ,ckpk)E~ = 0, 1 is a coherent set of frequencies.

xkZ

xkaO

PROBLEMS ON A-PERIODIC TRIGONOMETRIC SUMS

LEMMA 8:

If y ET,y' E

and y # y', then sup JxY(x)- xy.(x)l > xeG

115

43.

Proof: Let 6 = y - y'; then, for each x in G, (xY(x)- xY,(x)( I~a(x)- 1I. Since 6 # 0, the subgroup x,(G) of T is not the trivial subgroup (1). Hence there is an x in G such that Jx,(x) - 11 > J3. Now to the proof of Proposition 3. Let V be a neighborhocd of 0 in such that sup Ix,(x) - 1I < 1/(C 43). If S = y - y' E V, we have =

r

KxV

sup I~a(x)- 11 = sup Ixy,(x) - xy(x)I > l / ( J 3 C), which contradicts the K

K

definition of V. We shall say that A is uniformly discrete to express the conclusion of Proposition 3. If G is discrete, 7 is compact and each coherent set of frequencies A contained in T is finite. Hence coherent sets of frequencies are interesting when G is neither compact nor discrete.

10. Coherent sets of frequencies and restriction algebras

9. Coherent sets of frequencies in 1.c.a. groups The definition of a coherent set of frequencies will be given for the case of a general 1.c.a. group. DEFINITION 4: Let G be a 1.c.a. group, T the dual group of G and A a subset of A is a coherent set of frequencies if there exist a positive real number C and a compact subset K of G such that, for each trigonometric sum P(x) = xlen c r ~ ~ ( we x ) have

r.

(1) sup IPI G

< Csup IPI. K

If(1) holds we say that the pair (K, C) is suitable for A. Example: If G is compact, each subset A of has this ploperty. In the general case, each harmonious subset A of T has this property. But there seems to be no hope, even if A is relatively dense in of obtaining a characterization ofA as we did for harmonious sets.

r

r,

PROPOSITION 3 : LetA c I' be a coherent set of frequencies. There exists a neighborhood V of 0 in such that, if 2 EA, A' EA and 2 f: A', then 2' - A 4 v. The proof depends on the following lemma:

r

10.1. For each closed subset A of r , two Banach algebras A(A) and B(A) can be constructed: A(A) is the quotient algebra of the group algebra L1(G) by the closed ideal I(A) of L1(G) defined by f E I(A) if its Fourier transform vanishes on A ; B(A) is defined in the same way with the measure algebra M(G) in place of L1(G). We have the following surprising characterization of coherent sets of frequencies: A is a coherent set of frequencies if and only ifA is discrete and B(A) is the multiplier algebra of A(A). We first give some new definitions.

r,

DEFINITION 5 : Let E be a closed subset of the dual group of G. The Banach algebra A(E) is dejined by the following two conditions: (1) p, belongs to A(E) if there exists an f E L1(G) such that p, = f o n E; (2) llv/IA(E) = inf 11 f Ill, where the infimum is taken with respect to those f E L1(G), such that p, = .f on E. DEFINITION 6: Let E be a closed subset ofr. The Banach algebra B(E) is dejined by the following two conditions: (3) p, belongs to B(E) if there exists a bounded complex-Radon measure ,u E M(G) such that p, = , ii on E; (4) IIpIIBcE>= inf Ilpil, where llpll is the 'total variation' of p. DEFINITION 7: Let E be a closed subset of A P (G) is defined by

T.The subspace APE (G) of

116

PROBLEMS O N A-PERIODIC TRIGONOMETRIC SUMS

PROBLEMS O N A-PERIODIC TRIGONOMETRIC SUMS

(5) f E APE (G) if the spectrum of the almost-periodic function f is contained in E and the subspace L,"(G) is defned by (6) f E Lz(G) if the spectrum of the bounded Lebesgue-measurablefunction p is contained in E. Remarks: The Banach spaces APE (G) and B(E) can be put in duality. I f f E APE (G) is a finite trigonometric sum f(x) = CysE ayXy(x) and if p = ,L on E, we define n

117

(b) (a): The proof depends on the following construction of linear forms over the space APA (G). Let x E Hom (r,, T) be any homomorphism from the additive group I' (with the discrete topology) to T; for each f E A P (G), f(x) is defined by the following rule: iff is a finite trigonometric sum f(x) = zy,ayX, (x), f(x) is the corresponding ~ ~ m ~ ~ Kronecker's theorem shows that, on the finite spectrum A off, the complex numbers ~ ( y )y, EA, can be approximated by restrictions to A of continuous characters on T ; hence (f(x)I < sup If 1 and the linear form G

f +f(x) can be extended by continuity to the whole of AP (G). The following lemma will now be used:

We get I(f, p)l < Il f 11, llvllBcE,and, by Theorem I, the linear form f + (f, p) can be extended to the whole of APE (G).

10.2. Duality and coherent sets of frequencies It may be asked whether B(E) is the dual space of APE (G). The answer is surprising: THEOREM VI: Let G be a metrizable and separable 1.c.a. group andA a closed subset of the dual group F. The following two properties of A are equivalent: (a) A is a coherent set of frequencies; (b) each continuous linear form L on the space of all almost periodic functions P whose spectra lie in A can be written L(P) = f G P(x) dp (x).for a suitable complex valued bounded Radon measure p on G. Before proving Theorem VI some remarks should b: made. If for each positive 7 in 10, 1[ there exists a compact subset K of G such that the pair (K, 7) is suitable forA, then the dual space of APA(G) is isometric to B(A), as the following proof shows. In all other cases, the isomorphism between B(A) and the dual space of APn (G) is not an isometry. 10.3. Proof of Theorem VZ

A coherent set of frequenciesn is discrete in r;hence A is closed. (a) * (b) : Let (K, C) be suitable for A and let 8 be the closed subspace of %(K), whose elements are the restrictions g to K of all f in APA(G). A continuous linear form I on APA(G) defines a continuous linear form L on 6. The Hahn-Banach theorem shows that there is a measure p carried by K such that, for each f E APA(G), l(f) = f G f dp and (b) follows. The norm of p satisfies the inequality llpll < Clllll.

LEMMA 9: Let G be a metrizable and separable 1.c.a. group, r the dual group andA a subset ofr. Assume thatA is not a coherent set of frequencies. Then for each positive E, there exist a sequence (Pk)k>Oof trigonometric sums whose frequencies belong to A and a x in Hom (T,, T) such that (a) sup IPkl = 1 ; G

(b) on each compact subset K of G, sup lPkl+ 0, as k + +a; K

(c) IPk(x)I > 1 - E for each k 2 0. Before proving the lemma, we show why it implies Theorem VI. Assume that the linear form f +f(x), defined on APA (G) by the element x of Hom (T,, T) found in Lemma 9, can be written f (x) = fGf ( - x) dp (x) with p E M(G). Replacing f by Pkand using Lebesgue's dominated convergence theorem, we get f G P,(-x) dp (x) + 0, while IPk(x)I > 1 - &. This is impossible and Theorem VI is proved.

10.4. We return to the proof of Lemma 9. Let (K,),,, be an increasing sequence of compact subsets of G whose union is the whole group G. A sequence (Pk)k>Oof trigonometric sums on G whose frequencies belong to A and a sequence (xk)kgOof elements of G will be constructed by induction in such a way that sup lPk(= 1 G

(4.5)

~

~

118

PROBLEMS O N A-PERIODIC TRIGONOMETRIC SUMS

PROBLEMS ON A-PERIODIC TRIGONOMETRIC SUMS

We assume that x , , ..., x,, P,, ..., P, have been chosen and we now define x,,, and P,, , noting that the choice of x , and P, is easy. Since A is not a coherent set of frequencies, there exists a sequence ( Q J k a oof trigonometric sums whose frequencies belong to A such that sup IQ,J = 1 a r d sup lQ,l + 0 as k -> +a.The trigonometric sum

,

G

Kk

+ P, is an almost periodic function, as is evzry trigonoS, = P, + metric sum; so is Re [S,] and there exists a compact subset L , of G such that each translate s + L , contains an x satisfying Re [S,(x)] 3 sup Re [S,] - ~ 2 - ~ -3' k - F (2-I + + 2 - , + 2-k-2). G

Let m = m , be a sufficiently large integer that IQrn(x)I< l / ( k + 1 ) on K,,, - L,. Now regarding m as fixed, there exists an element s = s, in G such that IQm(s,)I > 1 We can multiply Q, by a constant complex number of absolute value 1 , which does not affect the other pro> 1 - d-,-'. NOWs, + Lk contains an perties of Q,, to get Re [Qm(sk)] element x k + , of G such that

119

11. Other definitions of coherent sets of frequencies

11.1. We recall that Hom (A,, T ) is the group of restrictions to A of all homomorphisms of the discretized group r i n t o the circle group T . Then thefollowing two properties of a closedsubsetA of r a r e equivalent: (a) A is a coherent set of frequencies; (b) Hom (A,, T ) is contained in B(A). Let A be a coherent set of frequencies. Each x E Hom (A,, T ) defines a linear form f +f ( ~ )on APA (G) and there is a ,u E M(G) such that f01> = jGf ( - X ) dp ( x ); this implies &(A) = ~ ( 1for) each A in A. If A is not a coherent set of frequencies, Lemma 9 gives a x E Hom (A,, T ) , which is not in B(A). 11.2. Let A be a closed subset of r.We say that a function y : A + C is a multiplier of A(A) if for each f E A(A), yf (the usual product) also lies in A(A).

r,

Putting P,+,(x)

=

Q,, ( x - x,,,

THEOREM V I I : Ij2l is a discrete subset of the following two properties of A are equivalent: (a) A is a coherent set of frequencies; (b) the multiplier space of A(A) is B(A). The proof of this result depends on the following interesting lemma:

+ s,), we have

and

The induction is thus complete. The inequalities (4.7) imply IPj(xk)l 2 1 - E for each k > 0 and each j less than k. Following Section 1 of Chapter 11, we regard G as a part of G" = Hom ( r d ,T ) ; let C , be the closure in G" of the tail-set {x,, x,, , .. .). Sii?ce G" is compact, there exists a 31 in G" belonging to all the C , , k > 0. Using the continuity of Pj for each fixed j, we get IP,(x)I 3 1 - F ; since x is independent of j, Lemma 9 is proved.

,

LEMMA 10: Let A be a discrete subset of a 1.c.a.grouprandlet f : A + C be afunction such that f(A) = 0 for all A in A with the exception of at most a cAf @)I,this supremum jinite number of A E A . Then Il f IIA(A) = sup being taken over all Jinite complex sequences (c,),,, such that P(x) = C n X A ( x ) satisjie~llPllrn < 1. Proof of Lemma 10: Let a = 11 f llAcA, and let b the right-hand side of the hoped-for equality. The inequality b < a is obvious. For each F > 0, there is an F in L1(G) such that llFlll < a F and P = f on A. Hence c ~ f (= 4 JG P( -x)F(x)dximpIie~ l & e ~ c ~( f4 < I1 PII, IIFII 1 < a + F. Thus b < a + E for all positive F , which grves b < a. We want to get the reverse inequality. Let E be any positive real number and g an element ofL1(G) with the following three properties: $(A) = 1 if A E A and f(A) # 0 , g has compact support and llgll < 1 F ([15],p. 53, Theorem 2.6.8). Let E be the finite set of all A in A for which $(A) Z 0. Theorem V I applied to the harmonious set E gives 11 f l l ~ ( ~<) b. This means that, for each positive E , there is an element ,u of M(G) such that ll,ull < b + F and

LA

+

,

+

121

PROBLEMS ON A-PERIODIC TRIGONOMETRIC SUMS

PROBLEMS ON A-PERIODIC TRIGONOMETRIC SUMS

,d = f on E. The convolution F = p * g belongs to L1(G); the norm

Proof: Let kj be an element of L1(G) such that kj = 1 on a neighbourhood of Ej and kJ = 0 on a neighbourhood of the union of the remaining E k , k # j , l
120

of F in L1(G) is less than (b + E) (1 + E) and f! = f on A. Hence a d (b + E) (1 + E ) for all positive E and Lemma 10 is proved. Now Theorem VII can be proved. The space of multipliers of A(A), for a discrete A, is exactly the dual space of APA (G) and our theorem is a corollary of Theorem VI. 12. Other relations between A(E), B(E) and L g

12.1. Theorem V I shows that B(E) is only exceptionally the dual space of APE(G). However, it is interesting to seek other relations of duality between the spaces A(E), B(E) and L;(G).

PROPOSITION 4: Let G be a 1.c.a. group,r the dual group and E, F two closed subsets of T such that E is contained in the interior of F For each f in L,"(G) and each p, in the restriction algebra B(F) a g in L g can be defined by 2 = g,? which furthermore satisJies llgll, d IIg, I 1 B ~ F )Ilf 11 ,. Proof: Let E be any positive real number. There exists a complex valued bounded Radon measure p on G such that ,d = g, on F and the total variation of p, llpll < (1 + E) 11~11~(~). Let g = f * p. This definition is independent of the choice o f p : if@ = B on F, then p * f = v * f since B is contained in the interior ofF. We get llgll d llpll Il f 11 ,d (1 E) Ilg, 11 s(~) (1 f 11,. Since g is independent of E > 0, Proposition 4 is proved.

+

COROLLARY 1: The notation is that of Proposition 4. Assume that E is compact. Then, for each f E LZ(G), there exists a complex valued bounded continuousfunction g on G such that f = g almost everywhere with respect to the Haar measure of G. Proof: There is an element k of L1(G) such that k = 1 on a neighborhood of E. Take g = f * k COROLLARY 2: The notation is that of corollary 1 and E is compact. For each f E L;(G) and each g, in B(F), a scalar product off and g, is given by f, g,> = j, f(-x) dp ( x ) , where ,d = g, on F.

<

3 : Let f be a complex-valued bounded continuousfunction on COROLLARY u Enof n disjoint compact subsets G whose spectrum is the union El u + .fn, where the fj are complexof r. Then f can be written f = f, + ualued bounded continuousfunctions on G and where the spectrum of euchfj , where C depends only on is contained in Ej. Moreover Il.fjl/,< C 11 f 11 , El, En. . . a ,

12.2. We have the following results on the algebras A(E) and BCE).

5 : If g, belongs to B(E) and has compact support, then g, PROPOSITION belongs to = IIvII B ~ E ) . and IIYIIA(E) Proof: The inequality I I v I I ~ ( ~ ) d I1911A(E) is immediate from the definition of these norms. To get the reverse inequality, let p, for each positive E, be a complex-valued bounded Radon measure on G such that fi = g, on E and llpll d (1 + E) Ilg,llscE).Let k be an element of L1(G) such that 11 k 11 d 1 + E and k = 1 on the support of g,. Then h = p * k belongs to L1(G), t%= g, on E and llhlll < (1 E ) IIg,llBcE). ~ Hence II~llacm< IIvIIB~E). PROPOSITION 6: The subalgebra A(E) of B(E) can be defined as the closure in B(E) of the subspace of compactly supported elements. Ifg, E A(E), then IIvIIA(E) = IblI B(E)Proof: The subset of all those f in L1(G) whose Fourier transforms have compact support is dense in L1(G) ([15], th 2.6.6, p. 51). Proposition 6 now follows immediately from Proposition 5.

+

13. Slowly-perturbed trigonometric sums 13.1. Intuitive motivation

Consider the motion of the earth around the sun; the position of the centre of gravity of the earth at time t is given by a continuous function g,: R + R3. To a first approximation, g, is a periodic function whose period T is one year and whose Fourier series is C?: a, exp 2kni (t/T). More precise observations show that the a, are not constants but functions of time whose periods are very great compared with T (secular perturbation). What is the quality of the approximation of these perturbed trigonometric sums y(t) = ak(t)exp2kni (t/T) by the corresponding periodic functions g, (to, t) = ak(t,)exp2kni(t/T); it is to be expectedthat the error term will be quite small in a neighbourhood of to. To state this problem in the context of general groups, the set ZT-' of frequencies used in the periodic case must be replaced by a coherent setA of frequencies.

x?:x?:

PROBLEMS O N A-PERIODIC TRIGONOMETRIC SUMS

Surprisingly, in order to solve our problem, it is necessary to know the behaviour of p, at infinity; in the real case, an apriori estimate of the type Ip,(t)l < Cexp Itla, C > 0, 0 < ac < 1, is sufficient (Section 17) and, in the general case of 1.c.a. groups, it will be assumed that p, is bounded. If A is a coherent set of (real) frequencies and if ~ ( t = ) a,(t) exp 2niilt is a slowly perturbed trigonometric sum by which we mean that the spectrum of each a, lies in a small interval [ - c , E ] , where E depends only a,(to) exp 2niilt is a very good on A, it will be proved that p, (to, t) = approximation to p,(t). In fact Ip,(t) - p, (to, t)l < A It - tol sup (Ip,(x)

CAE,

CAE,,

R

where A depends only on A, E and a.

Let G be a 1.c.a. group, F the dual group and A a subset of l'.We are interested in slowly-perturbed trigonometric sums of the type f,(x) x,(x), where the sum is finite and the f, are functions with slow variations. To be more precise, the following definition will be needed: DEFINITION 8: Let G be a 1.c.a. group, K a compact subset of G and E a positive real number. A complex valued bounded continuousfunction f on G s) - f(x)l < 11,. is (K, &)-slowijfor each x in G andeach s in K, I f(x In other words, each s in K is an &-almostperiod off. The following proposition provides examples of (K, &)-slowfunctions:

+

PROPOSITION 7: Let G be a l.c.a, group, r the dual group, E a closed subset of 'l and 6 a real number in 10, l[. Then each s in G such that sup IxY(s)- 11 < 6 is a 26-almost-period of each compex-valued bounded Y EE

continuous function f on G whose spectrum lies in E. Before giving the proof, we show why Proposition 7 gives examples of (K, &)-slowfunctions. Let U be a compact neighbourhood of 0 in r such that sup Ixy(x) - 1I < s/2. Then each complex-valued bounded contigKxU

uous function f on G whose spectrum lies in U is (K, &)-slow. Now to the proof. Let ol > 0 be an arbitrary positive real number sufficiently small that 6 + ac < 1 and let U be a compact neighbourhood of 0 in r , small enough to satisfy sup IxY(s)- 11 < a , where s is fixed YEU

throughout the proof. Let F = E

+ U.

123

1::

As in Lemma 4, we construct a discrete measure o = dl,&-,,,on G d , ~ ,(ns) = xY(s)- 1 for each such that lloll < 2 (6 a ) and b(y) = y in F. At this point, the following lemma is useful:

+

C?;

LEMMA 11: The function A , f defined on G by A, f (x) = f (x + s) - f(x) is equal to f * o. Proof: The Fourier transforms of A f and f * o are respectively (xY(s) - 1)fand bf. Since b = 2, - 1 on a neighbourhood of the support of the distribution 3: we have the required equality. Returning to Proposition 7, we have l l Af 11, < lloll llf 11 < 2 (6 + a) I ] f ]Ico Since this inequality holds for each a > 0, we obtain IlA f 11 < 2611f 11 ., Iff, is defined on G byf,(x) = f(x - s), the spectrum o f f , lies in E and we can apply the preceding proof tof, instead off. We get Ilf, - f 11, < 2611 f 11, and Proposition 7 is proved. Slowfamilies: DEFINITION 9 :Let G be a 1.c.a. group, K a compact subset of G, and E a positive real number. We say that a family (f,),EA of complexvalued bounded continuous functions on G is a (K, &)-slowfamily if, for every family (cJrEA of complex numbers of which all but afinite number are 0, the finite sum c, f, is (K, &)-slow. Example : Let U be a compact neighbourhood of 0 in the dual group r such that sup IxY(x)- 11 < ~ / 2 Then . each family (f,),,, of complex-

CAEA

Kx U

valued bounded continuous functions whose spectra lie in U is a (K, s)slow family.

13.3. Approximation of slowly-perturbed trigonometric sums

A fundamental result can now be stated: THEOREM VIII: Let G be a 1.c.a.group, l' the dual group andA a subset The following two properties ofA are equivalent: (a) A is a coherent set of frequencies; (b) There exist a compact subset K of G, a positive real number E and a constant C such that, for eachfinite (K, &)-slowfamily (fA),€, and each xo in G, the pure trigonometric sum f (x, , x) = fn(xo) XJX) is a good apfA(x)xI(x) proximation to the rlowly-perturbedtrigonometricsumf (x) =CnEA in the following sense:

of

r.

CnsA

124

PROBLEMS ON A-PERIODIC

TRIGONOMETRIC SUMS

PROBLEMS ON A-PERIODIC

Before proving this result, some remarks should be made. A finite family (fn)rEAis a family all of whose terms are the zero-function with at most a finite number of exceptions; (4.8) means that the approximation off by f(xo, x) is good in the neighbourhood x, K of x, and the estimates given by (4.9) will play a fundamental role in the problem of spectral synthesis. Proof (a) => (b): Assume that the pair (K, c) is suitable for A, and let E so small that EC < 1. Let M = sup I f(x, xo)l over all x E G and x, E G. For each fixed u, x +f (x, u) is (K, &)-slow since the family (fn)AeAis (K, &)-slow;hence if x E y + K, I f(y, u) - f(x, u)l d EM. Applying this remark with y = x, and u = x, we get I f(x,, x) - f(x)l d EM whenever x E X, + K. Since the pair (K, c) is suitable forA, so is the pair (K + x,, c) and, for each x, fixed in G, we have sup I f(xo, x)l < c sup I f(x,, x)l <

+

XEG

c sup (f(x)( G

+

xexo K

+ CEMsince I f(xo, x)l d I f(x)l + I f(xo, x) - f(x)l.

Taking

thesupoverx,~G,wegetM< cIlfII, + c & M a n d Md c ( 1 -8~)-'IlfII,; (4.9) is satisfied if we take C = cll - EC. Substituting this estimate of M in ( f(x,, x) - f(x)l < EM, we get (4.8). (b) (a): IfA is not coherent, there exists a sequence (Pk)kSOof trigonometric sums whose frequencies belong to A such that 11 Pkllrn = 1 and Pk-, 0 uniformly on compact subsets of G. Let f be any (K, &)-slowfunction vanishing at infinity and such that f(0) = 1. For example, let U be a compact neighbourhood of 0 in such that sup Ix,(x) - 11 < 4 2 and put f = 9, where g belongs to L1(r), Kx U

g is carried by U and g(y) dy = 1. By Riemann-Lebesgue's theorem, f vanishes at infinity and Proposition 7 shows that f is (K, 8)-slow. Applying (4.9) with x, = 0 to the perturbed trigonometric sums fPk, we get 1 = IIPkllwd C 11 fPkllco + 0, as k -f +a.We thus have a contradiction and so A is a coherent set of frequencies.

13.4. Better results on the approximation of slowly-perturbed trigonometric sums Our next theorem is an improvement of Theorem VIII. The terminology of Theorem IX is the following: (a) G is a metrizable and separable 1.c.a. group; (b) is the dual group;

r

TRIGONOMETRIC SUMS

125

(c) A is a subset of l;' (d) U is a compact neighbourhood of 0 in r and for distinct 2 in A, the corresponding 1+ U are disjoint; (e) for each x in G, r(x) = sup Ix,(x) - 1I ; Yeu (f) by a suitable normalization of the metric on G, we may assume that r(x) d 1x1, where 1x1 is the distance from 0 to x. With this notation, we can state

THEOREM IX: The following two statements are equivalent: (a) A is a coherent set of frequencies; (b) There exist a compact neighbourhood U of 0 in r and a positive constant C such that, for each x, in G and each perturbed trigonometric sum f = ~ r E A ~ for x n which the spectrum of everyfA is contained in U, the corresponding pure trigonometric sumf(xo ,x) = CnGA fA(xo)xn(x) satisjies the two relations

Proof: The implication (b) * (a) can be proved just as in TheoremVIII. If (a) is satisfied, let (K, c) be suitable for A and let E be a positive real number so small that sc < 1. Let U be a compact neighbourhood of 0 in such that sup Ix,(x) - 11 d 612. Proposition 7 now shows that

r

UxK

(fJAEA is a (K, &)-slow family and Theorem VIII gives (4.11) with C = (1 - &c)-l. We shall prove (4.10) using a constant D instead of C. We put a = r (x - x,). If a 2 1, there is nothing to prove: (4.11) implies I fx,(x) -f(x)l < ( C + 1) Ilfll, d DaIlfll,, where D = C + 1. If a < 1, we fn(u) ~ ~ ( x ) . define, for each fixed x in G, a function g on G by g(u) = Proposition 7, applied with s = x - x,, gives Ig(x) - g(xo)l d 2allgll , < 2Co1 11 f 11,; (4.10) is satisfied with C = max ((1 - EC)-' + 1 , 2 (1 - &c)-l).

CIEn

DEFINITION 10: Let G be a 1.c.a. group,A a subset of the dualgrouprand U a compact subset ofr. Thepair (A, U ) is a harmonic discretization ofA if, for distinct 1 in A, the corresponding 1 + U are disjoint and if the conclusion of Theorem IX holds. It will be now shown that each harmonic discretization ofA gives rise to an isomorphism between A (A + U) and A (A x U).

126

PROBLEMS ON A-PERIODIC TRIGONOMETRIC SUMS

PROBLEMS ON A-PERIODIC TRIGONOMETRIC SUMS

14. Coherent sets of frequencies and isomorphisms

14.1. Let G be a 1.c.a. group, F the dual group and A a uniformly discrete subset of T. This means that there is a compact neighbourhood U of 0 in 1"such that for different A in A, the corresponding A + U, i! EA are pairwise disjoint. There is thus a homeomorphism Hfrom the subsetA + U o f r o n t o the subset A x U of r2given by the rule H(A + t) = (A, t) if A EA, t E U. This homeomorphism defines an isometry 3 between V, (A x U) and %?,(A + U): for each f i n 'iP, (A x U), 3 ( f ) = f o H. We recall that A (A U) is the restriction algebra to A U of all Fourier transforms of elements of L1(G), that B (A + U) is the restriction algebra to A + U of all Fourier transforms of all complex-valued bounded Radon measures p in M(G) and that the corresponding notations are used for A x U, where G is replaced by G2.

+

+

THEOREM X: Let G be a I.c.a, group, r the dual group andA a subset of

r.Then the following three properties ofA are equivalent:

(a) A is a coherent set of frequencies; (b) A is uniformly discrete and there exists a compact neighbourhood U of 0 in r with thefollowingproperty: for each elementf of A (A x U),f o H belongs to A (A U), where H is the homeomorphism from A + U onto A x U dejined above; (c) there is a constant Csuch that,for eachf in A (A x U), 11f 0 HIIA(A+U) Q Cll f llAtAx v) and such that for each g in B (A x U), g o H belongs to B(A + U, and IIg HIIB(A+u) Q CllgllB(~xu).

+

14.2. To prove Theorem X, we need to introduce some spaces of testfunctions. The subspace d of L1(G) is defined by the condition that f E 6 if the Fourier transform f o f f is carried by a compact subset of Thc inversion theorem ([15], th. 1.5.1, p. 21) then shows that j"is the Fourier transform of f(j'(x) = f (- x)); hence f E A(G). It is well known that d is dense in L1(G) and it follows from the Hahe-Banach theorem that & is dense in %?,(G). The norm off in d will be sup If 1 = 11 f 11 m .

r.

G

If Q is an open subset of I', d ( 9 ) is the subspace of all f E I whose spectra are compact subsets of 9. PROPOSITION 8 : Let 9 be an open subset of T and E the closure of O. Let f be a complex-valued continuousfunction on E. Ifa positive number b can be found such that, for each a E &(O), ISEf(y) &Y) dyl Q bllallm, then f belongs to B(E) a n d 11 f (1 BtE) < b.

127

Let L be the linear form defined on 6(Q) by L(a) = jEf(y) &) dy. The metric form of the Hahn-Banach theorem shows that L can be extended to a linear form on V,(G) with the same norm. Hence there exists a complex-valued Radon measure p on G such that llpll Q b and L(a) = a(-x) dp (x) for all a e d(Q). The equalities jEf(Y) &) dy = jr & )I &) dy for each ol E 8(Q) and the continuity off imply fi = on Q; hence ,ii= f on E and Il f llB(E) < b.

I,

14.3. We return to the proof of Theorem X, (a) => (c): We retain the previous notation and introduce some new notation : (K, c) is suitable for A; 0 <eand~< c 1; V is an open neighbourhood of 0 i n r such that sup IxY(x)- 1I < 812; VxK U is the closure of V; d (A x V) is the normed space of all complex-valued bounded continuous functions ol on G x G, which can be written as finite sums a (x, y) = an(x) xn(y), where anE B(V) for each A EA. If g E B (A x U) and ol E 8 (A x v), we define (g, a) to be C2.n jr iY(y)g (1, Y) 4 ; d (A + V) is the space d(Q) for 0 = A + V; each element a E 6 (A V) can be written as a finite sum an(x)xA(x),where o r n ~ d ( Vfor ) each2in A; the correspondingelement of d (A x V) dejined byCnsAan(x)xn(y)will be called &. If g E B (A + U ) and a E & (A V), (g, a) = , j g(y) &) dy : this is a definition of the scalar product. To get (c), we begin with a g E B (A x U); Proposition 8 will be applied toprovethatgoH~B(A U).Foreacha~g(A V ) , ~ E ~ ( A V)X and Theorem IX gives llkllrn< CI1orllrn. We clearly have (g 0 H, a) = (g, G). Hence I (b): Trivial. (b) => (a) : Any restriction algebra is semi-simple. The general theory of Banach algebras ([I I], th. 4.1, p. 213; or [15], th. D5, p. 263) shows that the homomorphism f -t f 0 H from A (A x U) to A (A f U) is contin-

ILEA

+

xlEA

+

+

+

+

128

PROBLEMS ON A-PERIODIC TRIGONOMETRIC SUMS

PROBLEMS ON A-PERIODIC TRIGONOMETRIC SUMS

uous. Hence Il f 0 H l i ~ U,~ d~ Cll x f llAcA+u,. Again using the duality between &(A U) and B (A U) and between 8 (A x U) and B(A x U), we get the inequality l $ l < Cllall m . Theorem IX now shows that A is a coherent set of frequencies.

+

+

15. Improvement on the preceding results when A is harmonious 15.1. In the applications, Theorems IX and X are useless: they do not give any information about the size of the compact neighbourhood U of 0 in It would be convenient to take U to be the largest possible. We shall see that this is almost the case when A is an harmonious set. Before giving any precise result, Theorem VIII will be restated in the following way :

r.

r

PROPOSITION 9: Let G be a metrizable and separable I.c.a. group, the dual group andA c a coherent set of frequencies. There exist a compact neighbourhood U of 0 in r and a linear operator L, whose domain and range will be speczjied in a moment, such that (a) Iff is a complex-valued bounded continuous function on G whose spectrum lies in A U, g = L(f) is an almost-periodic function whose spectrum lies in A; (b) L is continuous: there is a constant C such that sup Jgl d C sup If 1

r

+

G

G

(c) Ig(x) -f(x)l d Clxl Ilf 11,; (d) For every 1 in A, the spectrum of g lies in (1) whenever the spectrum off lies in 1+ U; (e) There is a real continuousfunction OJ on G, vanishing at infinity, such that for eachf, llgll m G Ilwf ll m e Before proving Proposition 9, a remark should be made. Each coherent set of frequencies is uniformly discrete: there is a compact neighbourhood U of 0 in T such that for distinct 1in A, the corresponding 1 + U are disjoint. Each complex valued bounded continuous function f on G whose spectrum is a compact subset of A U can be written CAen f,(x) x,(x), where the spectrum of eachf,, 1EA, lies in U. To start with, we show that (a), (b), (c), (d) and (e) hold for any complex valued bounded continuous function f on G whose spectra lie in A + U if they hold for slowly-perturbed finite trigonometric sums f,(x) x,(x) for which the spectrum of each& lies in U. Finally, (e) will be proved in this last case.

+

I,,

129

15.2. Reduction to slowly-perturbedfinite sums

Let (kj)j,l be a sequence of elements of L1(G) such that Ilkjll = 1, kj is compactly supported, and kj + 1 uniformly on each compact subset of ([15], th. 2.6.8, p. 53). Put f j = f * kj. Then (a) f j = f iXA;the sum is finite and the spectrum off j lies in U ; (b) llfillmd llf llm; (c) f i +f, as j + + co,uniformly on each compact subset of G. Therefore (wf-'), j 2 1, is a Cauchy sequence for the sup-norm over G and, if (e) holds for perturbed trigonometric sums, (gj),, is also a Cauchy sequence for the sup-norm over G. Hence gJ tends uniformly on G to an almost-periodic function g. We put g = L(f); (a) is satisfied and the verifications of (b) and (c) are trivial.

r

xien

15.3. The case of slowly-perturbed trigonometric sums and the conclusion of the proof of Proposition 9

Let V be a compact neighbourhood of 0 in T contained in the interior of U, where U is defined by Theorem IX, and let W be a compact neighbourhood of 0 in sufficiently small to satisfy V + W c U . Let a be any function in the test-class d introduced in Section 14 such hat a(0) = 1 and whose spectrum is contained in W.

r

LEMMA 12: For each element f of LW(G),the Fourier transform of the product af is the function ;I* f i n A(r). Proof: af belongs to L1(G) since a does, and the Fourier transform of af belongs to A(T). We put a = & and to compute the convolution of the distributionfwith a,we test on a function /I in A ( 0 . Then jr (f * a ) /I dy = Jr(& * P)fdy: &(y) = n(-y) and the equality is the definition of convolution. By the definition ofA j, (& * /3)fdy = f G a$f dx. Hence the Fourier transform of af is h * f. Returning to Proposition 9, let o(f) be the spectrum off. Then the specA trigonotrum of af is contained in o(f) + W. If f = ~ l E fnXlisaperturbed metric sum and if the spectrum ofeach f,iscontainedin V, ~f= CnsAafnxl is a perturbed trigonometric sum and the spectrum of each af, is contained in U. Theorem IX shows that the pure trigonometric sum g = afA(O)x, satisfies SUP lgl G C SUP laf l and Idx) (af ) (XIl f ~ ( 0 )xn(x) =

CAE,

C

G

G

< CJxl jlaf 11,. To get Ig(x) - f(x)( d Dlxl 11 f 11 m ,it suffices to show that la(x) - 11 < C,lxl. Since 2 has compact support, this last inequality is a corollary of Proposition 7.

130

PROBLEMS O N A-PERIODIC TRIGONOMETRIC SUMS

PROBLEMS O N A-PERIODIC TRIGONOMETRIC SUMS

15.4. A counterexample Even ifA is a set of integers and U = [O, I], it is hopelessly difficult to compute the supremum lo of positive 1 for which Theorem IX is true. This supremum can be different from the supremum I, of positive I for which the intervals [A,2 I] are pairwise disjoint. IfA is the set of all integers of the form Aj = IOj, j E Z and j # 2" for all n 2 0, together with all integers of the form AJ = 10 2" - 1, j = 2", then I, = 9 while the conclusion of Theorem IX is false with 1 = 1. To see why the conclusion of Theorem IX is false in this case, consider any continuous n / 5 periodic function f. The Fourier series off can be written f(x) xtz c(j) exp ilOjx and the spectrum off is contained in A + [0, 11. If part (b) of Theorem IX was valid, there would be a continuous 2n-periodic function g(x) with the following two properties: g(x) ~ ( jexp ) iAjx and Ilgll < Cll f 11 a . But f - g is the lacunary 0 42") (exp i 1 0 . 2 " ~ - exp i (10 2" - 1) x). By a classical result series on Sidon sequences (Chapter VI, Theorem 11), there is an absolute constant 6 such that 11f - gll > 6 ( I n a o lc(2")I). This would imply 8 Ic(2")I) < (c + 1) Il f lla for any n/5-periodic function. But any sequence (x,).,~ in lZis the sequence of Fourier coefficients 42") of a n/5periodic function ([24], th. 7.1, ch. XII).

+

-

xzs

En,

-

(En,,

131

(a) iff is a complex-valued bounded continuousfunction whose spectrum lies in A + E, g = L(f ) is an almost-periodicfunction whose spectrum lies inA F ; (b) sup lgl < C sup 1 f 1; moreover there is a real continuous function w

+

G

G

on G, vanishing at injinity, such that sup Igl G

(c) for each x in G, Ig(x) - f(x)l

< 1x1 sup bf I;

< sup lwf 1; G

< Clxl sup G If 1 ;moreover Ig(x) - f(x)l

G

(d) for each 2 in A, if the spectrum off lies in 2 + E, the spectrum of g lies in 2 + F. Before proving Theorem XI, a remark should be made: since F is finite, the almost-periodic functions whose spectra lie in A + F are in a class of almost-periodic function because A + F is harmonious. Roughly speaking, the aim of theorem XI is to replace a bad function f by a regular one g without serious modification of the spectrum.

15.6. Summary of the proof The idea of the proof is to consider T as a dense subgroup of the Bohr compactification f of and, using a partition of unity on 17 by means of the regular functions, to divide A into a finite number of pieces AT for which Proposition 9 applied to the pair (E, A,*) will be true. We recall that the Bohr compactification f of is the dual group of Gd, where Gd is G with the discrete topology; A(P) - is the Banach algebra of complex valued continuous functions y on whose Fourier series are absolutely convergent. A continuous homomorphism h : f with a dense range is defined by the following rule: for each y ET,h(y) is the character X, on G. Then, if y E y 0 h is an almost periodic function on I' whose Fourier series is absolutely convergent.

r

r

15.5. A fundamental theorem With a very slight modification of the range of the operator L, the preceding difficulty disappears completely. Roughly speaking, Theorem XI states that each slowly-perturbed f,(x) x,(x) can be approximated by a pure trigonotrigonometric sum P,(x) x,(x), where the frequencies of each trigonometric metric sum sum PAbelong to a fixed finite subset F of the compact set E containing the spectrum of each f,. (We have introduced new frequencies to get a better approximation; A is harmonious and no restrictions have to be made on E.)

FA,, Inon

r

THEOREM XI: Let G be a separable and metrizable 1.c.a. group, the dual group andA a harmonious subset of T. For each compact subset E of I', there exist aJinite subset F of E, a positive constant C and a linear operator L, whose domain and range will be specijied in a moment such that

r

r-t

~(0,

15.7. A set F , If A is harmonious in I',let M(E), for each positive E, be the relatively dense subset of G defined by sup Ixn(x) - 1I < E. Let A, be the set of all A

y in

r such that Ixy(x) - I I < 28 for each positive E and each x in M(E).

Then A, is harmonious by Theorem I of Chapter I1 and (E - E) n A, = F , is finite. It is clear that F , = - F , and 0 E F1.Finally let be the set of all y in I' such that IxY(x)- 1I d s for all positive E and each x in

132

PROBLEMS O N A-PERIODIC TRIGONOMETRIC SUMS

PROBLEMS O N A-PERIODIC TRIGONOMETRIC SUMS

M ( E ) ;A is harmonious by Theorem I of Chapter 11. The above definition of F , seems artificial but Fl plays a fundamental role when A is mapped into the Bohr compactification f of I'. Bohr compactijication, U, A*, U*. Let h : I'-* 1; be the homomorphic injection of I' in f and let U be the closure of h(A) in f. The inverse image under h of the compact subset U o f f is again an harmonious subset of I' contained in as the following observation shows: For eachpositive E , each t E M ( E )and each x E U , we have Ix(t) - 1 I < E. The inequalities Ix(t) - I I < E , where t runs over M ( E )define a closed subset o f f containing h ( ~and ) hence containing U. For each y in h-'(U) we have Jxy(t)- 11 < E ( t E M(E))and h-'(U) c A. Let V be the compact subset h(E) o f f . The following lemma shows that the number of decompositions of an element x of f as a sum 5 g, 5 E U , g E V is finite; the finite set concerned here is precisely F,.

Ix(t) - g(t)l < & (resp. Ix(t) - g(t)l < &) for each t in T . Finally let cc, be the set of all g in f such that Ir(t) - 1) < for all t E T. As x runs over U , the family consisting of the corresponding a, is an open covering of U. There exist an integer n and a finite covering of U by o , , ..., on, where oj = w, for x = xj E U, 1 < j < n. We define U* = U + h(Fl), Uj* = U* n w,*, where a,*= w,* for x = xj and o,*is a compact subset of f'. Finally let A,* be the set of all A in I' such that h(A) E Uj*.The set A,* is contained in A* and the union U AS contains A.

+

+

LEMMA 13: If an element x of 1; can be written x = 6 q = 5' + qf, where 5 and 5' belong to U and g and g' belong to V, then 6 - 5' = g' - g belongs to h(Fl). Proof: By the definition of V, we have g' - g = h(y),where y E E - E. On the other hand, for each t E M(E),1Qt) - 11 < E and I['(t) - 11 < s, which imply I([ - 5') ( t ) - 11 < 28. Then IxY(t)- 11 < 2s (t E M(E))and y E E - E show that y E F 1 , since this is the definition of F, . In what follows, the compact subset U* = U + h(Fl) plays an important role; h-l(U*) = h-l(U) + F, is contained in the harmonious set A* = A -t F , (Chapter 11, Section 3, Theorem 11).

l<j
LEMMA 14: For each j such that 1 < j < n, Proposition 9, applied to the pair (A,*, E l ) is true. Since Aj* is contained in A*, we have IxA(x) - 1I < for each jl in A: and each x in M*. On the other hand, let a, be an element ofAj*, fixed once and for all. For every x and x' in wf and every t in T , we have Ix(t) - xj(t)l < +i and Ix1(t)- xj(t)l < &and hence Ix(t) - ~ ' ( t )< l &. This implies Ix, ( A - aj) - 11 ,< $i for a112 in A,* and all t in T (we recall that for y in I' and t in G, x,(y) = xy(t)). Finally we have Ix, ( A - aj) - 1 ) < & for every x in T + M* and every A in A,*. Proposition 7 shows that each x in T + M* is an +almost period of each almost-periodic function whose spectrum lies in A,* - a j ; hence the pair ( K , is suitable for Aj* - a j . The argument of Theorem 1X then shows that ( E l ,Aj* - aj) is a harmonic discretization of A,* - a j ; hence ( E l ,A;) is a harmonic discretization ofA7 (1 < j < n),

y)

15.9. Functions y j , 1 15.8. A partition of A

Let El be a compact subset of I' whose interior contains E + F,. We shall divide the harmonious set A into a finite number of pieces, denoted by A,*, 1 < j ,< n, in such a way that Proposition 9 is true for the pair (A,*, El), where A,* and El take the places of A and U respectively in Proposition 9. Let K be a compact neighbourhood of 0 in G such that, for each x in K and each y in E l , Ix,(x) - 11 < Let M* be the relatively dense subset of G defined by the condition that, for each x in M* and each A in A*, jxn(x) - 11 < &, a r d let T be a finite subset of G such that T + K + M* = G. For each x in P, let w, (resp. w,*)be the set of all g in I' such that,

5.

133

<j < n

Let V be the compact subset h(E) of f , U j the closure of U n wj in f ; then U j is contained in U and each x in U j satisfies the inequalities 1%-t) - xj(t)l < A, t E T. Let Q j , 1 < j 5 n, be an open subset of containing the compact subset U j + V of and small enough to satisfy the following condition: for any k in (1, .. .,n), Qj does not intersect Q k if U j + V does not intersect U k + V. The family SZ,, 1 < j < n, is an open covering of the compact subset U + V of f'. Let y,, ..., y,, be n continuous functions on R, whose Fourier series are absolutely convergent and which satisfy the two conditions: (I) the support of y j is contained in Q j ; + yn = 1 on a neighbourhood of U + V. (2) y 1 + The regularity of the self-adjoint algebra enables us to obtain such partitions of unity.

r

A(O

r-

PROBLEMS O N A-PERIODIC TRIGONOMETRIC S U M S

PROBLEMS O N A-PERIODIC TRIGONOMETRIC SUMS

We define pj = yj 0 h for each j in (1, . .., n). These pj are almostperiodic functions on f with absolutely convergent Fourier series; hence each g?, is the Fourier transform of a discrete bounded complex-valued measure oj on G. We have p1 + + p, = 1 on a neighbourhood of A + E and Wiener's theory (Chapter IV, Section 12) shows that each complex-valued bounded, continuous functionf on G whose spectrum lies inA+E,isasumf=f*o, +-..+f*o,.

Firstly, A - A* belongs to A* -Ag, which is harmonious, and to El - E; hence A - A* belongs to (El - E) n (A* - A*) which is a finite subset A of I'.We consider the translated sets E, of E by all a in A. For each s in E l , there exists an open neighbourhood J, of s small enough to satisfy the following conditions: for each LX in A, s belongs to E, if J, intersects E, and (J, - E) n (A* - A*) c A . We can find a finite covering of El by a finite number J, , . . ., J, of these J, corresponding to the elements sl , ..., s, of El.

134

0

.

.

LEMMA 15 : With the preceding notation, the spectrum of each f

1

< j < n, is contained in A: + El.

* aj,

The spectrum off * oj is the support of the Fourier transformfbj =fVj. To prove Lemma 15, it is sufficient to prove that if y belongs to A + E, which contains the support ofA and to the support of g?i, then y belongs t o n ) + El. The union U Uj is equal to Uand h (A + E) is contained in U + V. l$j$n

If we put x = h(y), we can find an integer k in (1, .. ., n} such that x E Uk + V. We can then write x = u + v, where u E Uk and v E V. Since y belongs to the support of y j , x belongs to the support of yj; hence x E Qj. If Uj + V does not intersect Uk + V, Qj does not intersect Uk + V; this is impossible since x belongs to these two subsets. Hence Uj + V intersects Uk + V. Lemma 13 shows that we can find elements 5 in Uj, 6' in Uk such that 6 - 6' E h(F1). Put u* = u + 5 - 5'. We shall prove that u* belongs to Uj*. Since u* belongs to U* = U + h(F1), it is sufficient to show that u* belongs to the closed set o) (U) = U* n oT and o) is defined like ojbut is 'three times larger'). But u and 6' belong to U,, and are therefore in the closure 13,of o k ;hence u - 5' is in - a, = I3 - ID.Sincel belongs to o j , u* belongs to o). Finally, by hypothesis, x belongs to h(r); so does each v in V = h(E) and u as 6 - E' are in h(I'). Hence the element u* of Uj* can be written u* = h(A*), where A* is in AT. Let us put 5' - 5 = h(f), f E F , , and v = h(s), s E E. The decomposition x = u* + 5' - l + v gives y = A* + f + s ; hence y belongs toll; + F, + E, which is contained inA7 + El.

15.10. Afinite set F The introduction of the finite set F and of a suitable partition of El allows to describe what happens if for some A* in A* and 2 in A, the translated set A* + El intersects 2 + E.

135

DEFINITION 11: With the preceding notation, F is the finite set of all dzflerences sk - a, 1 < k < q, a E A, which belong to E. We can state LEMMA 16: For each A* EA*, each k in (1, ..., q} and each A in A , if 2" + J, intersects A + E, then A* + skbelongs to A + F. We define elements C1, . .., [,of A(T)by the following two conditions: Cl + ... C, = 1 on a neighbourhood of El and the support of each Ck is contained in J,.

+

+

We define on the subset Aj* El of I',a complex-valued continuous function ZjPkby ZjVk(t)= Ck(s)if t = A* + s, where A* EA) and s E El. Theorem X shows that LEMMA 17: There is a complex-valued bounded Radon measure p j S k whose Fourier transform restricted toA7 + El is Zj,k. Operator L: Proposition 9 is true for the pair (AT, El). We can find a linear operator Lj,k, 1 < j < n, 1 < k < q, mapping complex-valued bounded continuous functions f on G whose spectrum lies in A) + J; (Jj, = El n J,) into almost-period functions Lj,,(f) whose spectra lie in A,* + sk;Lj,k has the following properties: (3) sup ILj,k(f < sup If 1 ; G

G

- f ( ~ > l< clxl Ilfllm; (5) if the spectrum off is contained in A* E l , A* E Aj*, the spectrum of Lj,k(f) is A* sk; (6) there is a real continuous function q,, on G vanishing at infinity such that IILj,k(f)II rn < I l o j , k f ll m . We now consider an f whose spectrum is contained in A + E. We define J;., 1 < j < n, by J;. = f * oj; the spectrum of fj is a subset of (4) I(Lj,k(f>>

+

+

136

137

PROBLEMS ON A-PERIODIC TRIGONOMETRIC SUMS

PROBLEMS ON A-PERIODIC TRIGONOMETRIC SUMS

A: + F, + E, which is contained in the interior of A,* + El. For this f,,, and hence reason we can define&,, to bef, * pjSk.We have f, =

(a) (b): If 8 is a Pisot number, A, is harmonious and Theorem XI1 is a weak version of theorem XI. (b) +-(a): The proof is based on the existence of a compact set E of real numbers with the following properties: (1) the vector space X generated by E over Q has Lebesgue measure 0; (2) 0 belongs to E; (3) there exists a complex valued continuous function pl on R vanishing at infinity, whose spectrum is contained in E and such that ~ ( 0 = ) 1. The existence of such a set will be proved in Chapter VIII, Section 10.

z

l
Ef,,,

f= where the sum is over all pairs ( j , k) such that 1 < j < n and l
16.2. Summary of the proof

Let Me be the set of all finite sums

C

pkOk,where pk E Z. Multiplying

k>O

E by a suitable real number if necessary, we can assume that 0 is the only element of Me lying in the vector space generated by E over Q. In this case the vector space of almost-periodic functions whose spectra lie in A + E has remarkable properties and (b) implies the conclusion of Proposition 9. We then use the existence of a continuous function on R, vanishing at infinity whose spectrum lies in E, to show that the validity of (b) implies that A, is a coherent set of frequencies. Hence 8 is a Pisot number.

+

16.3. Details of the proof

16. A new characterization of Pisot numbers

Let X be the vector space generated by E over Q. Reduction to the case Me nX = (0). Let Y be the set of all real numbers y such that yM, n X # (0). Then Y is contained in U 2-lX; since Me is

16.1. Let 8 be a real number greater than one and Ae the set of all finite sums &,ek, where &, = 0 or 1. It will be shown that Theorem XI applied to A, cannot hold if 8 is not a Pisot number.

zkTO

THEOREM XII: The following two properties of 8 are equivalent: (a) 8 is a Pisot number; (b) for each compact set E of real numbers there exist a constant C and a linear operator L, whose domain and range will be given in a moment, such that (1) iff is a complex-valued bounded continuousfunction whose spectrum lies in A, + E, g = L(f ) is an almost-periodicfunction whose spectrum lies in A, + E; moreover Ilgllm < Cll f / I m and g(0) = f(0); (2) if the spectrum off lies in 2 + E (2 EA), the spectrum of g also lies in 2 E.

+

AeMe

countable, Y has Lebesgue measure zero and so there is a real number y # 0 such that yM, n X = (0). If were place E by y - l E , then M, n X = (0) and the other properties of E are preserved. 16.4. Almost-periodicfunctions whose spectra lie in A,

The compact sets 2 + E, LEA, are pairwise disjoint since M, n X = (0). A trigonometric sum P whose frequencies belong to A + Ecan be written PAXI, where the spectrum of each PAis contained in E. For eachreal number x, the mapping H, from A + E to T defined by H, (A + t) = exp 27dixA is a weak character on A + E. Hence P(H,) is defined by

znen t

+E

138

PROBLEMS O N A-PERIODIC TRIGONOMETRIC SUMS

znEA P,(O)~,(x)and IP(H,)I < 11 PI], .The trigonometric sum Q(x) P(HJ satisfies IIQII, < llPll ,and the linear mapping T defined by T ( P )

PROBLEMS O N A-PERIODIC TRIGONOMETRIC SUMS

139

=

Q can be extended to the space of all almost-periodic functions whose spectra lie in A E. =

+

16.5. Let y be the complex-valued continuous function on R vanishing at infinity, whose spectrum is contained in E, and such that q(0) = 1. We apply (b) to the perturbed trigonometric sum f = Py, where P is any trigonometric sum whose frequencies belong to A; we get an almost periodic function g with the following properties: c a x ~g, = ZAG, g,~,, where each ga is an almost-perio(i) if P = dic function whose spectrum lies in E; iii) g m = c,; ( W Ikll m < CIIPII ,. We now define Q to be T(g); then Q(x) = CnEA dAxn,d, = g,(O) and IIQII, < IIT(g)ll,. Then Q = P and we get IIPll, < CIIPyll,. Since p, vanishes at infinity on R, A is a coherent set of frequencies.

I,,,

17.2. We end this chapter by describing the connection between the local approximation property given by statement (3) of Theorem XIII and the arithmetical structure of A. Let 6' > 1 be a real number, A the set of all finite sums Ekek, E , = 0 or 1 , and A , the set of the 2k sums t j O J , cj = 0 or 1.

ZE-l

PROPOSITION 10: If 0 is not a Pisot number and E is a positive real number, statement (3) of Theorem XI11 does not hold for A. Proof: In fact we shall prove the stronger result that (3) fails in any neighbourhood of to. Let a : R -+ R be a continuous function, vanishing at infinity, not identically 0, such that a(0) = 0, and whose spectrum is contained in [ - E , E]. Let Pk(t) = 2-k exp 2 d t . Then sup R la(t)Pk(t)l + 0 as k

-+

-+

[ l , + co] be an even log-concave function such that +

* log w(x)

sup la(h) P, ( t

THEOREM XIII: Let w be as above. Then, for each coherent set of frequencies A there exist a positive E and a constant C > 0 such that (1) i f f : R -+ C is a continuous function which is 0 (w(t))at injinity and whose Fourier series is a,(t) exp 2 d t andif the spectrum of each a, is contained in [ - E , E ] ,then, for each real t o ,f ( t o , t ) = a,(to) exp 2 d t is almost periodic; (2) SUP lf(t0 t)l < C SUP If(x)/w ( x - toll ;

x,,,

2

R

(3) If(t0, t ) - f(t)l

R

< C It

+ h)] = a(h)

-+

0 as k +

+ h). If (3) were + co;

C-h,hl

if a(h) # 0 this cannot happen and, since a is an entire function, we can find such an h as close to 0 as we like.

dx

is finite. For instance, w(x) = exp lxla is such a function if a < 1.

xaen

+ co as was proved in Section 5.2. Let

for some h # 0. The corresponding f (0, t ) is a(h) P, ( t true for to = 0 on the interval [-h, h], we should get

17. Slowly-perturbed trigonometric sums in the real case

17.1. Let w : R

zkSO

- tol SUP lf(x)lw (x - toll. R

Proof: The Denjoy-Carleman's theorem ([I I], p. 114, th. 2.4) shows that for each positive E , there is a real continuous function a supported by [ - E , E ] ,whose integral is 1 and such that 2 = 0 (llw) at infinity. We now reproduce the proof of Proposition 9, to get Theorem XIII.

18. Notes The main result of this chapter was Theorem XI. In the real case, we considered the real line as a dense subgroup of a suitable compact group G in such a way that our discrete harmonious set A was contained in a thin compact subset of G . This idea has been used before by J. F. MCla in [65] and [66] to study the so-called 'interpolation sets for almost periodic functions'.

INTERLUDE

If G = R, r = R and A = Z, E(A) is the space of all continuous 1-periodic functions f : R -t C; E(A) is trivial on each compact subset V of 10, 1[. More interesting examples will be studied in Chapter V. If G = Q,, r = Q, and A is the subset of all mpn in Q, such that 0 < mpn < 1, where m and n are otherwise arbitrary rational integers, then E(A) is trivial on each compact subset V of Q, whose Haar measure is less than the measure of Z, (Theorem IX of Chapter V). Proof of (a) => (b); A is a relatively dense harmonious subset of r:for some compact subset K of r , A + K = r. Theorem I1 of Chapter IV shows that for each positive E there is a relatively dense set of &-almost periods for all f in E(A). The proof of the last statement of (b) is an easy adaptation of the proof of part (b) in Theorem IX of Chapter V. For this reason, the reader is referred to Chapter V, Section 8.7. Proof of (b) =. (a): The existence, for each positive s, of an a priori complete set M(E) of &-almostperiods for all f in E(A) implies that A is harmonious. To prove that A is relatively dense, from which (a) will follow, we use the following lemma :

INTERLUDE

One of the most fascinating problems in harmonic analysis is to find connections between the additive properties of a set A of frequencies and the functional properties of the corresponding space of almost periodic functions whose spectra are contained in A. ForA a finite set of real numbers independent over Q, we obtained the following connection in Chapter I: we can rapidly predict the size of trigonometric sums whose frequencies belong to A if and only if A is strongly independent over Z (Chapter I, Theorem IX). In the general context of 1.c.a. groups, we can also get striking results. We first recall some definitions: (1) A subset A of a 1.c.a. group G is discrete if each point 1 of A is isolated in A for the relative topology; (2) A subset A of G is relatively dense if there exists a compact subset K of G such that each translate of K intersects A ; (3) A set E of bounded continuous functionsf : G -+ C has a complete set of a priori &-almostperiods if there exists a relatively dense subset M(E) of G such that, for each z in M(E) and each f in E, sup f ( x + z) G - f ( x ) l Q E SUP If I ;

LEMMA 1 : For each n > 1, let A, be the set of a l l y in r whose distance from 0 exceeds n. Then for any h :I' -+ C in A(r), the norm of h in the restriction algebra A(A,) tends to 0. Proof: Obvious, since the compactly supported h in A(r) are dense in A(r). We return to the implication (b) (a). If E(A) is trivial on a compact neighbourhood V of 0 in G, a theorem of Banach applied to the surjective restriction operator T: E(A) -+ C(V) yields a constant C such that, for each complex valued continuous g, in C(V), there exists an f in E(A) such that sup If 1 Q C sup Ig,I and f lv = 9.

I

G

(4) A set E of bounded continuous functions f : G -+ C is locally trivial at 0 if there exists a compact neighbourhood V of 0 such that each continuous function g : V 4 C is the restriction to V of a function f in E. In this case we say that E is trivial on V. With these definitions we can state the following: THEOREM I: Let G be a metrizable and separable 1.c.a. group, I'the dual group, A a subset of I', and E(A) the space of all almost periodic functions f:G -+ C whose spectra lie in A. The following two properties are equivalent: (a) A is discrete, A is relatively dense in P and there is ajinite subset F of r such that A - A is contained in A F ; (b) for each positive s, there is a complete set of a priori s-almost periods for all f in E(A) and E(A) is locally trivial. Before giving the proof, some examples should be noted.

+

141

G

,'

V

It follows from duality that for each complex valued Radon measure p carried by V and each bounded complex valued Radon measure v, the equalities $(A) = $(A) for all 2 EA imply Ilpll < Cllvll. Let h be the Fourier-transform of an integrable function I, not identically zero, which is carried by V; then h E A(r). IfA is not relatively dense in G, there is for each n > 1, a yn in r whose &stance from A exceeds n. Let dp, = y,(x) I(x) dx. Lemma 1 shows that there is a sequence v, of complex Radon measures such that llvnll -+ 0 and ,il,(l)= Jn(2) for all A inA. But IIpnll = 111111and this contradicts llpnll Q Cllvnll.

SPECIAL SERIES

CHAPTER V

that, for every a

E 9'

10 SPECIAL T R I G O N O M E T R I C SERIES (COMPLEX METHODS)

(COMPLEX METHODS)

satisfying

+ 1x1)" y'k'(x)l < C,

0

< k < n,

(5.1)

we have IT(a)l < 1. The Fourier transformation a + & is an isomorphism of Y and the Fourier-transform of each T i n 9' is defined by the rule ( T , &) = (F, a ) . An element T of Y' is supported by [0, oo[ if for each a E Y which is 0 in a neighbourhood of [0, a [ , we have ( T , a ) = 0. From now on we shall c o n h e our attention to those T E Y' supported by [0, + a ] .

+

In the real and the p-adic case, precise and interesting results about coherent sets of frequencies need complex methods. In Section 1 we define the Laplace or the complex Fourier-transform of a distribution with compact support. The Paley-Wiener theorem gives precise information on the set of zeros of such a transform (Sections 2 and 3). In Section 9, we present the theory of mean-periodic functions on the line. Studying coherent sets of frequencies is equivalent to investigating the behaviour at infinity of all mean periodic functions whose spectrum is a given set of real numbers. Trigonometric series whose frequencies belong to the set of all entire parts of kB, where k 2 1 and 13 is an irrational number greater than one have quite fascinating properties which are studied in Section 6. But perhaps the most beautiful results concern the trigonometric series al exp 2ni2x, where 2 = mpn, defined on thep-adic field Q, by f ( x ) = CAE, for rational integers m and n such that 0 < mpn < 1. These series have almost all the properties of 2mperiodic functions on the line and other better ones (Section 8). The harmonious set A and its homothetical copies pkA, k 2 1 , are of outstanding importance in the atomization of distributions in Q,, just as arithmetic progressions are used to discretize problems on the real line.

143

+

1.2. The domain of dejnition of the Laplace transform

+

Let p = u iu be a complex number. The function t -+ e-"' does not belong to 9 , but i f a is an indefinitely diflerentiable function which is 1 on [- 1 , co[ and 0 on ] - oo, -21, then a ( t ) e-"' belongs to Y for each p E C such that u = Wep > 0. We can define ( T , e-"') to be ( T , a ( t ) e-P') whenever T is supported by [0, a [ . From now on, we write e-"' T ( t ) dt instead of ( T , e-"'). The Laplace transform can be defined on a larger space of distributions. For instance consider the function exp J7, t 2 0. For each p E C whose real part is positive, the integral j2 exp ( - p t &) dt converges and defines the Laplace transform of this function. Let 9 be the space of all complex valued indefinitely differentiable functions on the line with compact support. A distribution is a linear form T over 9 such that for each m Z 1 there is an integer n > 0 and a constant C > 0 with the property that for all y in 9 supported by [ - m , m ] the inequalities sup Ip(j)(t)J< C, 0 < j < n, imply IT(y)l < 1.

+

+

+

1-m,ml

If this is the case we write T E 9'. The domain of definition of the Laplace transform is the subspace 9of 9' consisting of all distributions T carried by [0, oo] such that, for each E > 0, e-et Tbelongs to 9'.

+

1. The Laplace transform 1.1. The space of tempered distributions

Following the notation of [21], Y is the Frechet space of all complex valued functions a on R all of whose derivatives, together with a itself, decay rapidly at infinity: for each j Z 0 and each k 2- 0, (1 Ixl)j y'k' ( x ) + 0 as 1x1 + + oo. The dual space Y' is the space of tempered distributions. For each T E Y' there is an integer n and a constant C > 0 such

+

142

1.3. Properties of the Laplace transform PROPOSITION 1 :For all T i n 9,theLaplace transform @(p)= :j e-"'T(t) dt is holomorphic in the open half-plane Wep > 0. Proof: Let 0 < E < Wep. Then e-pr = e-"e-'p-e'r , e-er T ( t ) E 9' and W e ( p - E ) > 0. The integral is thus well defined.

S E 9"; if S E ~we, apply the preceding argument to S ( t ) exp ( - s t / 2 ) with s/2 instead of E. The first part of Theorem I is thus proved. In the second half of Theorem I, the unknown distribution Twill be a limit of convolutions T * f as f runs over a sequence f , of indefinitely differentiable functions of the form f k ( t ) = k g ( k t ) , where g is supported by [O, 11 and j g(t) dt = 1. From now on, we write f instead off,. Let F(p) be the Laplace transform off. The Laplace transform of ~ k f is pkF(p). Now the Laplace transform of a complex valued continuous function carried by a compact subset of [0, + a [ is bounded. Hence 1pkF(p)l < c k .

To prove that @(p)is holomorphic, we apply the same trick and we are thereby reduced to examining only the case where T belongs to 9". If Wep, > 0, then with a defined as above, ~ ( t(exp ) ( - p t ) - exp (-pot)/ p - po tends to -ta ( t ) exp ( - p o t ) in the topology of 9'.Hence @(p)is differentiable over the field C at the point po and @'(p) is the Laplace transform of - t T (t). THEOREM I : Let T be a distribution carried by [0,+ co[ such that, for each positive E , e-,'T is a tempered distribution. Then the Laplace transform @(p)of T is a holomorphic function in the open halfplane with the following property: there is an integer n 2 0 such thatfor each positive s, we can find a constant C , such that (5.2) I@(p)l < C,lpln for Bep 2 s. Conversely each holomorphic function in the half-plane 9 e p > 0 satisfying (5.2) is the Laplace transform of a unique distribution T carried by [0,i-a[ such that e-" T belongs to 9" for each positive E. The proof of the first half of Theorem I begins with a lemma; here T is any distribution carried by [0, a[.

91ep20

The p~oduct@,(p) = F(p) @(p)has the following behaviour at infinity: for each positive E > 0 and each n > 0, Idil(p)l < C ( n , E ) IpI-" when Wep 2 E.

+

+

LEMMA 1: There exist an integer n >, 0, a continuous complex oalued function f carried by a compact subset of [0, + a [ and a distribution S carried by [l, + co [ such that T = D n ( f ) + S where Dn is the nth (distributional) derivative. Proof of the lemma: Let B be any indefinitely differentiable function with compact support, which is equal to 1 on a neighbourhood of [0,11. We write T = BT + ( 1 - B) T . The distribution /IT is carried by a compact set. Hence there exist an integer n 2 0 and a continuous function g such that g(t) = 0 if t < 0, g(t) = c if t >, to and PT = Dn(g).Let y be any indefinitely differentiable function with compact support, which is equal to 1 on a neighbourhood of [0,t o ] We define f = yg; then D n ( f ) = Dn(g) on a neighbourhood of [0,to] and T = D n ( f ) + S is the required decomposition of T. Proof of the first part of Theorem I: Let @, be the Laplace transform off and@, the Laplace transform of S. Then @, is an entire function ofp, bounded on Bep 2 0. The Laplace transform of D n ( f ) ispn@,(p)and satisfies (5.2). On the other hand, let a be an indefinitely differentiable function which is 0 if t < and 1 if t 2 2. Then for each s > 0 and each n 2 0, there is a constant c = c (n, E ) such that each function q ( t ) = ~ ( texp ) -pt (Wep >, E ) satisfies (5.1). This implies I<S, q)l < C, if

+

LEMMA 2 : For each real t , the integral ( 1 1 2 4 t): @, ( u iv) et("+'")dv T f ( t )exists and is independent of u > 0.This integral vanishes i f t < 0. Proof of the lemma: Let u, < u, be two such u. We apply Cauchy's theorem on the rectangle whose vertices are (u, , -L), (u, , -L), (u, ,L ) and ( u l , L). We get =

i

i StL@, (u,

+ iv) ef(u"i") dv - i j!,

Dl (u,

+ iv) e"U1'i"' dv

The behaviour of @, at infinity ensures that the last two integrals tend to 0 as L tends to infinity. I f t < 0 and u 2 1 , IJ+; @, (u + iv) et(u+iu)do] < C ( 2 , 1) j?," lu ivl-, dv, which tends obviously to 0 as u tends to + a .

+

LEMMA 3 : For eachjixed u > 0,e-'"Tf ( t ) E Y . The Laplace transform of T f is @, . Proof: The function e-'"Tf ( t ) is the inverse Fourier transform of @, ( u iv). To get estimates of the derivatives of @, , we apply Cauchy's formulas on the rectangle with vertices ( 4 2 , v/2), (3u / 2 , ~ / 2 )( ,3 ~ 1 23012) , and ( 4 2 , 3 ~ 1 2 ) :the inequalities I@, (u/2 iv)l < C'(u, n) Ivl-" and I@, ( 3 ~ 1 2 iu)l < CV(u,n) Ivl-" imply that, for each fixed u, each derivative of @, (u iv) with respect to v decays rapidly at infinity. Hence v -+ @, ( u iv) belongs to Y and taking the inverse Fourier transform,

+

+ + +

+

The Fourier-transform of edtUT f ( t )is ( u + iu); the Laplace transform of T , is therefore @, . We now replace f by f,(t) = k g ( k t ) and we write T k instead of Tf,. LEMMA 4: AS k tends to inJinity, T k tends to a distribution T carried by [O, + co [ in the sense that for each indeJinitely diferentiable function h with compact support, ( T k ,h ) + ( T , h ) as k -+ + co. Proof: Let E be any positive real number. The Fourier transform of e - , ' ~ , is @, ( E + iv) = @ ( E iv) yk( E + iv), where yk is the Laplace transform of k g (kt). We remark that lYkl < A throughout the halfplane Weu >/ 0 for some A independent of k and that Yk-+ 1 uniformly on compact subsets of this half-plane. The function eeth ( t ) is the Fourier transform of a function r(v) in Y . @ ( E + iv) Y k ( &+ iv)r(v)dv Hence ( T k , h ) = ( e - " ~ ~ eeth) ,, = + ST: @ ( E + iv) r(v) dv as k + co (we have I@ ( E iu)I lr(v)I dv < + co since @ grows slowly at infinity and r decays rapidly at infinity and Lebesgue's dominated convergence theorem shows that the limit exists). If a distribution T is defined on R by the statement that the inverse Fourier transform of @ ( s + iv) is e-" T ( t ) , T would appear to depend on E. But the preceding argument gives, lim ( T k , h ) = ( T , h). Hence T k++m does not depend on E , Tis carried by [0, + oo [ and the Laplace transform of T is @.

+

+

(b) there exist an integer n 2 0 and a constant C such that IF (iy)l < C (1 lyl)". Conversely each entire function F(z) satisfying (a) and (b) is the Laplace transform of a distribution carried by [-I, I].

+

2.2. Proof of the first part of Theorem 11 If S is carried by [-I, I ] , we can find, for each positive E, two continuous functions f and g carried by [-I, I + E ] and an integer n 2 0 such that S = D n ( f ) + g. The proof is the same as that of Lemma 1. Let @(z) be the Laplace transform off, G that of g. Then F(z) = zn@( z ) + G(z). @(z) = j'_:,exp ( - z t ) f ( t ) dt implies I@(z)l < j'_:'exp ( - x t ) I f(t)l dt < C(E)exp (1 + E )1x1. The same inequality holds for G(z). Hence

+

2. The Paley-Wiener theorem

2.1. We list a number of corollaries of theorem I which are interesting in themselves. But first a definition is needed: 1: A complex valued.function f dejined in the complex plane DEFINITION is of exponential type I if, for each positive E , there exists a constant C(E) such that I f(z)l < C(s) exp [(I + E )lzl] for all z. A similar definition can be given for functions of a real variable. We say that f is of exponential type iff is of exponential type I for some I in 10, co [.

+

THEOREM 11: Let S be a distribution carried by [-I, I]. Then the Laplace transform F(z) of S is an entire function in the complex plane with the following two properties: (a) F is of exponential type I;

which implies (a) and (b). 2.3. To prove the converse, let F(z) be an entire function satisfying the inequality (5.3) for each positive E. Then exp [-(I + E )z ] F (z) is holomorphic in 9 e z > 0 and is bounded by C(E)(1 + IzIn). Theorem I shows that exp [-(I + E )z ]F ( z ) is the Laplace transform of a distribution T carried by [0, + co [. Hence F(z) is the Laplace transform of a distribution S carried by [- 1 - s, a[.In the same way, it can be proved that S is carried by ] - oo,I + E ] . The uniqueness of S shows that it is carried by

+

[-I,

4.

It remains to show that (a) and (b) imply (5.3). We use the following lemma of Phragmen-Lindelof : LEMMA 5: Let Y be a closed sector in the plane whose opening is less than n radians, and let f ( z ) be an holomorphic function on a neighbourhood of Y. I f f is of exponential type on Y and if( f(z)l < 1 on the frontier of Y, then ( f1 < 1 on the whole of Y. Proof: Since this statement is invariant under rotation, we can assume that Y is defined by -a < Arg z < a < 4 2 . Iff tends to 0 when z E Y and lz] -+ co, the maximum modulus principle gives I f(z)l < 1 for all z in Y. If f ( z ) does not tend to zero at infinity, we apply the preceding argument to f,(z) = exp (-EZ") f(z), where a > 1 and a a < 4 2 . We have lexp ( - E Z " ) ~= exp ( - E W e (za))< exp ( - s lzla cos aa). Hence I f,(z)l < 1 for each frontier point of Y andf , tends to zero when z E Yand lzI -, + co.

+

The preceding argument gives I fE(z)l < 1 for all positive E and all z in Y. Letting E tend to 0 , we get I f(z)l < 1. COROLLARY 1 : I f F is holomorphic in a neighbourhood of x 2 0,y 2 0 , i f F is of exponential type and ifJF(x)l < C exp Ix, IF (iy)l < C (1 (yl)", then (F ( x iy)l < C (1 Jzl)"exp lx. Proof: We apply Lemma 5 to f ( z ) = C-l ( z + i)-" F(z) exp -1z. The proof of Paley-Wiener's theorem is complete. We have proved the following improvement of Paley-Wiener's theorem. If F is an entire function of exponential type (not specified), if there exist an integer n 2 0 and a constant A such that IF (iy)l < A (1 Jyl)",and if for some positive I there is for each positive 8 a constant C , such that IF(x)J < C , exp (I + E ) 1x1, then F is the Laplace transform of a distribution carried by [-I, I].

+

+

n(r) = k nl(r). Once Theorem IS1 is proved for F , ,this relation at once gives the result for F. We apply Jensen's formula to F , to get that for each positive E and each R 2 1

+

+

+

Since this inequality holds for each positive E , Theorem I11 is proved. Remark: If n(r)/r tends to a limit D as r tends to infinity, so does R - l n(r)/r dr and Theorem 111 gives D < 1. In general, let D = l& n(r)/r. Then i E i R-l n(r)/r dr 2 D r-r + m

and Theorem 111 gives D

3. Repartition of roots of entire function of exponential type In all of what follows, F will be an entire function of exponential type I, so that for each positive E , there is a constant A(&) such that JF(z)l < A(&)exp (1 + E )(zI for any complex number z. Let a,, ...,a,, . . . be the sequence of all roots of F, ordered in such a way that la,l < la,+,l, and with each root occurring in the sequence (a,),, a number of times equal to its multiplicity.

,

DEFINITION 2: For each positiue real number r, n(r) is the number of roots of F in the closed disc lzl < r; alternatively, n(r) is the sup of n > 1 such that la,] < r. The function n(r) is defined on [0, +a[, is integervalued and is a step function. THEOREM 111: If E is an entire function of exponential type I, then lim sup R-I 1: n(r)/r dr < 1. We start with the well-known Jensen's formula: if F(z) is holomorphic on a neighbourhood of Jzl < R and if 0 < [all < la21 < ... < la,l is the sequence of roots of F, then dr o

r

=

S:'

2n

log IF ( ~ e ' ~do. )l

The proof is given in [16],th. 4.1, p. 181. We return to Theorem 111. I f F ( 0 ) = 0 , F(z) = zkF, (z),where F,(O)# 0 and if nl(r) is the number of roots of F , in the closed disc lzl < r, we have

r

o

Hence

< 1.

R-+m

,,,

COROLLARY 2: Let (A,)-, +, be an increasing sequence of real numbers such that lim I k / k = I / D exists. If 1 < n D , ecery distribution S Ikl-r+m

carried by [ - I , I ] whose Fourier transform canishes at all the A,, k is the zero distribution. Proof: Apply the Paley-Wiener theorem and Theorem 111.

E

Z

4. Bernstein's inequality THEOREM IV: Let g, be a complex valued bounded continuous function on the line whose spectrum1 Iies in [ - I , 11, or equivalently, a complex valued bounded continuous function on the line which is the restriction to R of an entire function of exponential type I. Then sup 19'1 < I sup Jq1. R

R

In particular for each pair of real numbers x' and xu, Ipl(x') - y(xV)l < 2 Ix" - x'l sup 191. R

Proof: The equivalence of the two definitions of is a consequence of Paley-Wiener's theorem. Let T / 4 be any real number greater than I and p ( t ) the T-periodic function defined by p(t) = 4T-'t if It( < T / 4 and p(t) = - 4 T - l t 2 if

+

In Sections 4 and 5, Fourier-transforms are defined by fix) =

e-lx'f(t)

dt to be

TI4 < t < 3T/4;p is the well-known sawtooth function of physics. Let q(t) = p ( t - T / 4 ) ;the function q is positive-definite and has an absolutely convergent Fourier series: q(t) = c, cos (2knt/T),ck 2 0, ck = 1. Hencep(t) = dk cxp (2knit/T),where Id,l = 1. I f o i s thediscrete measure dk6 ( x 2 k n T - l ) where 6 is the Dirac measure, the11b = p . Letg, be anycomplexvalucd boul:dedcontinuous functionon theline whose Fourier-transform S = is carried by [-I, I]. The Fourier-transform of y' is i t s ( t ) = i ( T / 4 ) p ( t )S(t). Hence g,' = i (T/4)a * g, and llg,'Il < (TI41 llall IIg,llm = (T/4) IIyllm. Since this inequality is true for each T / 4 > I, Bernstein's inequality is proved.

1,"

z?: +

x+:

x:

6

5. Other inequalities of Bernstein type THEOREM V: Let (Ak)- m < ,< + be an increasing sequence of real numbers such that lim (Aj+ - S ) / k = 1 ID uniformly over j E Z . For each

,

0 < A,,, < ( k + 1) L if k 2 0 and k L < A,,, < 0 if k < 0. Cantor's diagonal process yields an infinite increasing sequence m ( j ) of integers such that + A; for each k 2 0. For all k E Z , 0 < 6 < Am,,+ < L ; hence 0 < 6 < A;+, - A; < L and the set A' of all A;, - lm,k \ - co < k < + co is a regular set of real numbers. The preceding proof shows that if A has a realization A,, k E Z , such that lim (Aj+, - Aj)/k = 1/D uniformly over j E Z , so does k-+m

A': for each E > 0, there is an integer k ( ~ such ) that k 2 k ( ~implies ) (D-I - E ) k < Aj+, - Aj < (D-I + E ) k. But we can find an integer I depending only on m such that for each k in Z , A,,, = A,,, - x , . Hence for each j in Z , each m 2 1 and each k 2 k ( ~ )we , have (D-I - E ) k < - Am,j < (D-I E ) k . Passing to the limit, we get (D-I - E ) k < A;+, - AS < (D-I + E ) k for j~ Z and k 2 k ( ~ )which , is the required inequality.

+

k-+m

I < nD, there is a positive constant C such that for any complex valued bounded continuous function g, whose spectrum lies in [-I, I ] , we have ldAk)l. 191 < R

-m
5.2. To prove Theorem V, let E be the Banach space of complex valued continuously differentiable functions on [-21,211 and V the convex set of all distributions S carried by [-I, I ] whose Fourier-transforms 3 satisfy the inequality sup < 1. Then V is a compact subset of the dual space R

5.1. Proof of Theorem V Theorem V will be dednced from Corollary 1 of Theorem 111 and standard compactness arguments that we now describe. A set A of real numbers is called regular if there exists a 6 > 0 such that each interval of length 6 contains at most one point ofA and if there exists an L > 0 such that each interval of length L contains at least one point ofA. A realization cf A is a presentation of A as an increasing sequence A,, - co < k < co, of real numbers. We have 6 < A,,, - A, < L and all other realizations of A are of the form ,uk= A,-,, ( - a < k < + a ) A . sequence A,, m 2 1 , of regular sets of real numbers tends to a regular set A if there is a realization A,,,, - co < k < $ co of eachA, and a realization A,, - co < k < co, ofA such that A,,, + A, for each k . We do not require uniformity of convergence with respect to k .

+

+

LEMMA 6: Let A be a regular set of real numbers and x,, m 2 1, a sequence of real numbers. There exist a regular setAf of real numbers and a subsequence xk of x , such thatA - xh tends ton' when m tends to infinity. Proof: Let A,,, be a realization of A - x , such that, A,,, 2 0 for k 2 0 and A,,, < Ofork < 0. We haveil,,, < L,A,,-, 2 -Landhence

E* of E with respect to the weak-star topology a (E*, E). For each real t the mapping from Q to the complex field defined by S + $(t) is continuous. If a sequence (S,),, of distributions carried by [-I, I ] exists such that sup 131 , = 1 while sup lgm(Ak)l< llm, let x , be a real number

,

R

-m
+m

such that I~,(x,)I 2 5. Let T,, be the distribution e-'""" T ; T , belongs to V, J(T,, 1)1 2 4 and IF, (Ak - xm)l < l / k . We can find an increasing sequence m j of integers such thatA - x,(, +A' and Trncj) + T' as j tends to infinity. Bernstein's theorem shows that for any two real numbers x' and x" and any S i n Q, 1$(xr) - 3(x1')l < Ixr - xtll. Applying this inequality to X' = A; EA' and x" = A,,, E A - x,, whose limit is A;, we get, writing m instead of m j , pm(A;)-+ 0 as m + + co ; hence Pt(A;) = 0 for all k in Z . The set A' of all 31; has a uniform density D and D < nl. Corollary 2 implies T' = 0. This is impossible siwe I(T, 1)1 = lim I(Tm, 1)1 2 3. m-

+m

Hence Theorem V is proved. For a better result, the reader is referred to [54].

6. Irrational numbers and special trigonometric series

6.1. Let 8 be an irrational number > 1 and, for each positive integer k , let [Bk]be the integer m defined by 0 < Bk - m < 1 ; m is the integral part of 8k. We define A(8) to be the set of all [Bk],k E Z. DEFINITION 3 : For each irrational number 8 > 1 , g ( 8 ) is the Banach space of all complex valued continuous I-periodic functions whose spectra lie in the set A(B)of all in tegralparts [ k e ] ,k E Z. The Fourier series o f f is a, exp 2ni [kB]t.

1 :

THEOREM VI (properties of V ( B ) ): (1) g ( 8 ) is a closed, translation invariant space of I-periodic functions on the line; (2) i f O is any open subset of [0, 11 of Lebesgue measure greater than 118, eachfunction in g ( 8 ) is completely deJned by its values on 52, as thefollowing properties show: (b) i f f isp-times continuously diferentiable on O ,f isp times continuously diferentiable throughout the real line; (c) iff is analytic on 52, f is analytic on the real line; (3) i f K i s any compact subset of [0, 11 ofLebesgue measure less than 118, any complex valued continuousfunction on K is the restriction to K of some function in V(8). A p-adic version of Theorem VI will be given in Theorem IX. Theorem VI fails if 8 is rational.

finite union of intervals U whose Lebesgue measure is greater than 118. Once (a) is proved for U, it is proved for 52. Let M be the set of all integers m in Z such that h(m) E U . For any sequence ( c ~ ) ~ of , , complex numbers, we put

With this terminology we can state two lemmas:

7: For any complex valued continuous function f defined on T , LEMMA lim %l {f ( h ( j ) ) ; m < j < m k ) = ( 1 1 2 ~ ) f(eie) dB, uniformly over

+

k++m

m

Sin

Z. Since 11/2nS?f(eie) dB1 < Ilfll, and I%R{f(h(j));m < j < m k)l < 11f 11 , it suffices to consider the dense subspace of finite trigonometric sums; by linearity it is sufficient to examine the case where f(eie) = elpe, p E Z . In this case an easy computation gives the result. E

+

LEMMA 8: For any complex valued Riemann integrable function g on T the same result holds. Proof: By linearity we can confine our attention to real g. For each positive E , there are two continuous functions f l and f , on T such that fl < g < f2 and ( 1 1 2 4 j? (f2 - f l ) (eie)dB < E. Hence

6.2. Sketch of the proof of (a) The proof is based on the following remark: under the injective homomorphism h from Z to T defined by h ( j ) = exp 2nij8-17 the image of Z is a dense subgroup of T , and A(8) is mapped onto a dense subset of the arc I of T defined by the parametric representation z = exp -2nit, 0 < t < 118. Iff in a function defined on T whose spectrum lies in A(8) its composition with this homomorphism is a function g defined on Z whose spectrum lies in I and we are reduced to Theorem V. 6.3. A computation of densities

Before going into the details, some remarks should be made. An open subset O of [O,l] whose Lebesgue measure is greater than 110 contains a

But the two extreme terms of this inequality differ from (1 1222) t)? f(eie) d8 by an error term not exceeding E (Lemma 7). Hence lim %R { g (h(j)); m

< j < m + k ) = (1127~)

k-++oo

g(e'e) dB uniformly on m

E Z.

LEMMA 9: The uniform density of the set M of integers is equal to the Lebesgue measure of U. (Note that the Lebesgue measure of the whole of T is 1.) Proof: We apply Lemma 8 to the characteristic function of U.

6.4. Proof of (a) To prove (a) for all f in V(I3), it suffices to prove (a) for finite trigonometric sums P(t) = ck exp 27ci [kO] t. Moreover, each open subset Q of [O, 11whose measure is greater than 118 contains a finite union U of closed intervals, the sum of whose lengths is greater than 1/13. If we prove (a) for U instead of 52, the inequality with Q will hold a fortiori. Let h : Z + T be the homomorphism defined by h(j) = exp 2nijI9- l. The dual homomorphism h* : Z + T is also defined by h*(j) = exp 2nijI9-l. Since h*(Z) is a dense subgroup H of T, we have sup IP( = sup IPI C0,ll H and sup 1PI = sup ]PI. U

UnH

LetM be the set ofallintegersj~Zsuchthat h*(j)~Uandlet,ubethe measure carried byT and assigning the mass ckto the point exp -2niI9-'[ke]. An easy computation gives P(j) = P (h(*j)). To get (a), we have to prove that sup I,G(j)l $ C sup I$(j)l for all finite linear combinations of Z

M

Dirac measures concentrated in points of the form exp -2niI3-I Eke]. We now make the following crucial observation: let I be the arc of T defined by the parametric representation z = exp - 2nit, 0 < t 6 1/13. Then h(A) is contained in I; the image of the set A(I3) under the homomorphism h lies in an interval. Writing [kO] = k0 - r where 0 < r < 1, we get 8-I [ke] = - 0-lr (mod I), which proves our remark. Using the local inverse of the momomorphism t + 2nit, we now replace I by the interval J = [- (I/@, 01; ,u becomes a measure v carried by - J and all that remains is to show that sup le(t)l $ C sup IB(t)(. ZnZ

ZnM

Theorem V shows that to get this inequality, it suffices to show that 27cM is a regular set whose uniform density is greater than 23-218. Lemma 9 shows that this is the case and (a) is proved. LetQ be an open subset of [0, 11 of Lebesgue measure greater than 110. If a function in V(I9) is zero on Q, this function is zero everywhere. If two functions of V(I9) agree on Q, they are equal everywhere.

6.5. Proof of (b) and (c) Let f be a function of V(I9) continuously differentiable on an open subset Q of [0, 11 whose Lebesgue measure is greater than 110. Let U be a finite union of closed intervals contained in Q the sum of whose lengths is greater than 118. On U , the family of functions g (x) = t - I [f(x t)

+

- f(x)] of V(I9) tends uniformly to f '(x) as t tends to 0. Theorem VI, (a), implies uniform convergence over the whole of R ; hencef is continuously diffcrentiable throughout R. Similarly, iff is k times continuously differentiable on Q and iff belongs to V(I9), then f is k times continuously differentiable on R and sup ( f ( k ) ( R < A sup If (k'(,where A is independent of k > 0. B

Assume that f is analytic on 9 acd let U be as above. The inequalities If (k'(x)l $ Ckk! on U imply I f(k'(x)) $ ACkk! on the whole of R and f is analytic throughout R.

6.6. Two remarks Remark 1: For each open subset Q of [0, 11 whose Lebesgue measure is greater than 1/19, there is a constant C such that for any function f in V(@, J: lf(x)I2 dx G CJf2 If(x)12 dx. Proof of this remark: Let U be a finite union of closed intervals of [0, I] such that, for a positive E , U + [-E, E]is contained in Q, and such that the Lebesgue measure of U is greater than 118. Let g be any complex valued continuous function whose support lies in [ - E , E] and such that llgllz = (r-&lg(t)I2 dt)lI2 < 1. For each f in V(I9), the convolution product f * g lies in g(I9) and, by part (a) of Theorem VI, for each real x,

Taking the sup with respect to g, we obtain, for each real x,

and

Remark 2: For each open subset SZ of [0, 11 whose Lebesgue measure is greater than 118, there is a constant C such that for any complex valued Radon measure ,u whose spectrum lies in A(I9), Il,ull $ C JB dlpl (t). The proof of this second remark is similar to that of the preceding one : the set U and the positive real number E are defined as above, the test function g is any continuous complex valued function supported by

158

SPECIAL SERIES (COMPLEX

METHODS)

(b) iff E gAis p-times continuously diferentiable on K, then f is p-times continuously differentiable throughout the real line; (c) iff is analytic on a neighbourhood of K, f is analytic throughout the real line; (d) if K is a compact set of real numbers whose measure is less than l/la/3 - olbl, (a) and (b) are false. Before proving this theorem, some comments should be made. If E is any positive real number and K is the union of the two intervals I, = [0, (a/3- abl - s] acd .Ym = [m - E, m + E ] ,then, even if m is very large and the two intervals are very far from each other, the values off on K determinef completely. On the other hand the knowledge off on I,, is not sufficient to reveal any substantial information. The role played by the very small and distant interval Y mis indeed curious. The set A of all finite sums ,Qk, E~ = 0 or 1, 0 = (1 + 2/5)/2 can also be defined as the set of all positive real numbers m + no = 1, m E Z, n E Z such that - 1 < m n% < 0 where 8 = (1 - &)/2. In this case 3 J?/10. the critical measure 1/lap - abl is

Ek,,

+

++

7.2. Computations of densities

To obtain a better understanding of the number lllaj3 - abl, the following lemma will be useful: LEMMA 11 :Each intervalof realnumbers of length Tcontains T lap - olbl + o(T) points ofA; o(T)/T tends to 0 uniformly with respect to the position of the interval as T tends to infinity. To prove this lemma, we in fact consider a more general situation which is of interest in itself; the corresponding result will be proved for a large class of harmonious and relatively dense sets of real numbers. Definition of models: Let n be an integer greater than 1 ; U , a compact subset of Rn-l; a,, ...,a,, n real numbers linearly indeperder~tover Q ; I, the linear form defined over Rnby I (x, , ..., x,) = a,x, + . + n,,x,; L,, ..., L,-, , n - 1 other linear forms such that L,, ...,L,-, , I constitute a basis of the dual of Rn; (L: ,...,L,*) is the isomorphism of Rnwhich is the adjoint of the inverse of the isomorphism L of Rn defined by x + (L,(x), . ..,L,- ,(x), Ilx)),

X E R " : for each x and y in Rn, L:(x) L,(y) + + L,*_l(x)~,-l(y) L,*(x) I(y) = x . y. We assume that the coefficients of L,* are linearly independent over Q. Under these assumptions, we construct a set of real numbers which is called a model (Chapter 11, Section 6); A is the set of all real1 = a,k, + a,k, for all k = (k,, ..., k,) in Znsuch that the element (Ll(k), ..., L,- ,(k)) of Rn-I belongs to U.

+

+

7.3. A Poisson formula with remainder term for a model (computation of densities) In the following theorem U andA are defined just as above; E is any positive real number and U, is the set of all x in Rn-l whose distance from U does not exceed E . Finally A, is defined using L,, ...,L,-,, I and U,. THEOREM VIII: Assume that U is Riemann integrable, i.e. that the frontier of U has Lebesgue measure zero. For each positive E there exists a complex measure p e such that (1) sup j=+ldlp,l(t) < + a ; -m<x<+m

(2) the Fourier transform Sp, of p, is also a measure; Sp, is equal to (x - 1) + R,(x), where B(x) is the unit mass concentrated the sum in (0); (3) the remainder term R,(x) has the form '&sAs\A@(A)6 (x - 1) where 0 <&(A) < 1; (4) lim 1 / 2jz;: ~ d IR,(x)l + 0 as E + 0, uniformly over all real x.

CnEAB

T++m

This means that the Fourier transform of p, is the sum of unit masses concentrated in each point ofiltogether with an error term. This error term consists of positive masses, less than one, concentrated in points of Ae\A1. The mean value of this error term is arbitrarily small as E tends to zero. Now to the proof. We start by defining a function a which will be extended to the whole ofA, by the rule a(1) = 1 if A E A. Let p, be a Cm function on Rn-l, which is 1 on U and 0 outside U,. If 1= alk, + + ank, we put k = (k,,...,k,) E Z", &(A) = y(L1(k), ...,L,-,(k)) and p is a CMfunction defined on Rn by p(x) = p, (L1(x), ..., L,- ,(x)). A, &(A) B (X - 1) where B(x) is the unit mass concentrated in Let v = (0) and let us compute @. Iff is any test function in 9,we have

XI

A,\A is the complement of A in A,.

The function /3 (x, , ..., x,) f (a,x, + -.. + a,x,) decays rapidly at infinity over RE; its Fourier transform Q (x,, . .., x,) = (det L)-, @ (L: , ..,L,*- ,)f(~,*)also decays rapidly at infinity. At this stage, the well-known Poisson formula will be applied. If e : Rn-, C belongs to Y and if 6 (y, , ..., y,) = JRnexp -2ni (x, y, + .. xnyn)e (xl, .. .,x,) dxl dx,, then e(k) = &k). In our case this equality gives

.

xz,

+

xzn

The last equality follows from the observation that (gp,) * y is an almost periodic function whose Fourier coefficient for the frequency x is +(x)pA -4. Letting c tend to zero we get Lemma 12. Let V be the closure of U,\U. We have dens (A,\A) < (det L)-I meas (U,IU) and since the measure of the frontier of U is 0, lirn dens A,\A E'O

= 0. Since the masses of Re lie in ]O,l[ and are concentrated in points of

A,\A, (4) follows. COROLLARY 3 : If U is Riemann integrable, lim (1/2T) Card ([x - T, where p is the atomic measure assigning the mass (det L)-' L ),: (k,, ..., k,) to the point L,* (k,, ..., k,). We have

$J

(L: , ...,

x

+ TI n A,)

T-+m

= (det L)-l meas U, uniformly over all real x.

7.4. Proof of Theorem VZZ

where the latter sum is extended over all k in Zn such that L,*(k) belongs to [x, x 1). Let 0 be the characteristic function of [0, 11. Then

+

I:"

d Ip(t)l

(det L)-l

=

C (@I zn

(LT , ...,L L l ) 0 (L,* - x).

Since the linear forms L? , ...,L: form a basis of the dual space of Rn, there exists a E E Rn such that L,*(t) = ... = L,*_,(E) = 0 and L,*([) = x. Let w be the function defined on Rn by co = (det L)-I (L:, .. ., L,*-l) 0(L,*). With this notation d Jp(t)l = o (k - 6). Since w decays rapidly at infinity, we have sup w (k - 8 < + CYJ and hence

CZn

xzn

sup J?'

d Ip(t)l <

xsR

+ co.

CER~

We now have to prove (4). We need the following notion : for any compact subset V of Rn-l, let A, be defined by (L, , ..., L,-, , I ) and V. We define dens A, to be lim (2T)-l sup (Card [x - T, x + TI nA,). xeR

T*+m

LEMMA 12: dens A, < (det L) - meas V. Proof: Let y be any indefinitely differentiable function defined on R, whose support lies in [0, I] and such that ji y(t) dt = 1. We construct, using V, a measure p, just as we did with U. Then, since R, is a positive measure, (2) gives

-

dens A,

<

J

lirn -

'

~ - + + m2T

(Sp,) ,-T

* y dx = p,(O) , uniformly over x E R.

Let h be the homomorphism from R to T2 defined by h(t) = (exp 2niat, exp 2niPt). To prove (a), it is sufficient to verify (a) when f(t) is a finite sum a, exp 2ni1t since these sums are dense in V, with respect to the topology of uniform convergence over the whole of R. Such a sum can be written P o h, where P is defined on T2 = R2/Z2 by P (x, y) = C(rn,n)eT am., ~ X 2ni P (mx + ny). For all (x,, yo) in R2 and all real t, we have

xAEn

P (x,

+ at, yo + bt) = C

ah,, exp 2ni (am

+ bn) t = g(t),

(m.n)~r

where a&,, = a,,, exp 2ni (mx, + ny,). This function g(t) is the Fourier transform of a measure carried by [- 1, I]. Let h' :R -t T2 be the affine map defined by h'(t) = (exp 2ni (x, + at), exp 2ni (yo + bt)). Since bla is irrational, hl(R) is dense in T2. Let A' be the set of all real1 such that h'(1) E L = h(K); thenn' is a model whose uniform density is lap - abl meas K and hl(A') is dense in h(K). Theorem V gives sup Ig(t)l < A sup Ig(t)l = A sup IPI. Since ht(R) tsR ten' L is dense in T2, sup (g(t)l = sup IPI. Since h(R) is dense in T2, sup I f(t)l Tz

t ER

tsR

= sup ]PI; h restricted to K is a homeomorphism from K to L and this T2

implies sup 1 f 1 K

=

sup IP(.Hence sup I f(t)l L

~ E R

< A sup If 1 and TheoremVII K

is proved. Property (a) implies that, for each Riemann-integrable compact set K of real numbers whose measure exceeds l/lap - abl, there exists a constant C such that, for each complex-valued almost-periodic function f

whose frequencies belong to A,

8.3. Proof of (a)

Landau proved in [54], th. 2 p. 39, that this last property implies meas K 2 dens A. This is not the case here. Another method of proving this type of statements will be given in the p-adic case.

Replacing K by a smaller compact subset if necessary, we may assume that K i s a finite union of disjoint open balls B j , where an open ball of Q, is a coset of the form xZ, + y, x E Q,, y E Q,, and that the measure of K still exceeds 1. Let A' be the set of real numbers defined by the rule: A EA' if A E A, and furthermore if A, regarded as an element of Q,, belongs to K. LEMMA 13: A' can be ordered in such a way as to become an increasing sequence (A;)- ,,,, +, of real numbers such that lim - jl;)/k k+ + m = 1 /IKI uniformly over j E Z. Proof of Lemma 13: An open ball of Q, can always be written B = xZ, + y, where x = pJ, j E Z and y E A,. Then B n A, considered as a set of real numbers, is a two-way infinite arithmetic progression of common difference ]XI,' = IBI-l and uniform density IBI. Lemma 13 follows from this observation. We return to the proof of Theorem IX. Let P be any finite sum A ar exp 2nixA ( x E Q,) and p the corresponding trigonometric sum defined on R by p ( t ) = a* exp 2nitA. The frequencies of H belong to 10, l[. Since A, is dense in Q, and also in R, we have sup IP( = sup IPI

8. Special trigonometric series on p-adic fields

8.I. Let Q, be the p-adic field, Z, the subgroup of p-adic integers and A, the ring of elements mpn, m E Z, n E Z, of Q,. By an abuse of notation we shall identify A, with a subring of Q. L e t A be the subset of Q, consisting of all mpnin A, such that 0 < mpn< 1. By Hensel's decomposition, eachp-adic number x can be written uniquely as x = A + r, where A EA , r E Z,. Hence each coset x + Z,, x E Q,, of the compact group Z, meetsA in a single point. The Haar measure, denoted by dx, is normalized on Q, by IZ,I = 1. For each x in Q,, we define exp 2nix to be equal to exp 2 n d if A E A, and x - A E Z,. With this notation, each continuous character on Q, can be written in the form x -t exp 2nixy.

xlE =

sup

=

sup

8.2. W e denote by V A the Banach space of all complex-valued almostperiodic functions on Q,, whose spectra lie in A. The set of all finite sums P(x) = CleAa, exp 2niAx is a dense subspace of g,.

I

1

THEOREM IX: Let K be a Riemann integrable compact subset of Q,. (a) Ifthe Haar measure I K ( of K exceeds 1, there is a constant A such that or each f in V A , sup J f l 4 A sup 1 f l i QP

sup I

~ Similarly, I. asA'

A,

QD

n K is dense in K, sup IPI = sup /PI

R

K

A'

Since A' is regular and has uniform density IKI, Theorem V

A'

gives A sup If31 2 sup IP"I and the proof is complete. A'

R

8.4. Necessary andsufJicient conditions for sup If ( QP

< A sup ( f1 K

To prove (c), we first give the following characterization of compact subsets K of Q, such that sup If ( and sup If 1 are equivalent norms for funcK R tions f in %'A.

K

(b) If the Haar measure IK( of K is less than 1, each complex-valued continuous function deJned on K is the restriction to K of a function in V A . In particular, there i~ a non-zero function in V A , which vanishes identically on Ki (c) The subgroup Z, of Q,, whose Haar measure is exactly 1, does not satisfy the conclusion of (a). For a more precise result, the reader is referred to [89].

=

An

I

I

xleA

I

PROPOSITION 2: Let A = A, n 10, 1[, let K be a compact subset of Q,, let A' = A, n K considered as a set of real numbers and let A be a positive constant. The following two statements are equivalent: (a) for each almost periodic .function f : Q, + C whose frequencies lies inA,supIfl 6 AsupIfl; K

QP

(b) for each bounded continuousfunction g, : R 4 C whose spectrum lies in [O, 11, sup 191 < 4' sup 1 ~ 1 . R

A'

where z is the measure supported by A, aiid assigning the mass a(t) to each t in A,.

Assuming, for the moment, that Proposition 2 is proved, let us see why statement (c) of Theorem IX follows. If K = Z,, A' = Z and ~ ( x = ) exp 2nix - 1 is such that the spectrum of v lies in [0, 11 and q~= 0 on Z while sup (TI = 2. For this reason 2,

LEMMA 15: With the notation of Proposition 2, the following two statements are equivalent: (a) SUP If l < A SUP If I;

R

does not satisfy the conclusion of (a) in Theorem IX. In the same way, it can be shown that if Kis the union of a finite number of disjoint open balls of Q, and if IKI = 1, then K does not satisfy the conclusion of (a) in Theorem IX.

QP

8.5. Proof of Proposition 2

Enen

The interest of Proposition 2 is that it establishes an equivalence between properties of functions on the p-adic field and properties of functions on the real line. The proof of Proposition 2 depends on a series of lemmas:

LEMMA 14: Let a : R + R be the function defined by a(t)

=

K

(b) for each t in A,, there exists a complex Radon measure p, carried by K, whose norm does not exceed A and such that ,&(A)= exp 2ni1t for each 1in A. In the proof it suffices to confine attention to the dense subspace of finite trigonometric sums whose frequencies belong to A. (a) => (b): Let d be the vector space of all restrictions to K of finite trigonometric sums P(x) = Cn exp 2niilx; the norm of P in d is sup If 1. If (a) holds, we can define a linear form L, over d for each t in A, K

by L,( f ) = f(t); the norm of L, does not exceed A. By the metric form of the Hahn-Banach theorem, there is a complex Radon measure p, carried by Ksuch that L,( f ) = jK f dp, for eachf in 8;testing onf(x) = exp 2nixt we get (b). (b) => (a): Let P be any complex valued trigonometric sum whose frequencies lie in A. Then (b) gives P(t) = j, P(-x) dp, (x), which implies IP(t)l < A sup IP(. Since A, is dense in Q, we have sup (PI = sup \PI

sup (0, 1

- 12t - I(); a is the Fourier transform of the function b : R+ R dejined by b(x) = (exp inx) (1 - cos nx)/(n2x2).Let o be the atomic measure carried by A, and assigning to eachpoint t in A, the mass b(t). Then the Fourier transform of o, in the sense of distributions of Q,, is also an atomic measure t, carried by A and assigning to each 1in A the mass a(3L). Remark: The measure o is not bounded, but an elementary computation gives lo1 (x + Z,) = 1 for each x in Q,. Proof: Let D c Q, x R be the discrete and relatively dense subgroup of all pairs (t, t), t E A,. The annihilator of D is the discrete subgroup D' of Q, x R of all pairs (-t, t), t E A,. A class Y of test functions can be defined on Q, in the following way; f : Q, -t C belongs to Y if there exist n disjoint open balls B, , . .., B, and n complex numbers a,, ..., a, such that f restricted to Bj equals aj and f

K

< A sup ]PI.

QP

K

LEMMA 16: With the notation of Lemma 15,for each t in A,, let v, be the restriction of p, to A, and let Q, be the remainder term, i.e. Q, = p, - v,; then for each 1in A, &(A) = $,(A). Proof: Let o be the atomic measure carried by A, and defined as in Lemma 14. The convolution product p, * o is well defined since one of the two terms has compact support; testing on functions in Y , we see that the Fourier transform of p, * o is kt&,the product of a regular function with an atomic measure. If 1E A , the mass of P,b concentrated in 1is a@) exp 2ni1t, where a(1) denotes the value of a on 1considered as a real number, and if il @ A ,the mass of P,b concentrated in 1 is 0. Hence $,8 = a@) exp 2ni1t d(x - A), where d(x) is the unit mass concentrated in 0. Let st be the unit mass concentrated in t; then p, * o = st * o since these two measures have the same Fourier-transform. But p, * o = v, * o + Q, * o and v, * G is carried by the group A,, while Q, * o is carried by

n

equals 0 on Q,\U Bj. The class Y is invariant under Fourier transforms. 1

To compute the Fourier transform 8 of o, we consider the integral jQp$(x) do (x) = '&f(x) b(y), where the sum is over all pairs (x, y) E D. ~EQ~,YER. The Poisson formula ([7], Vol. 2, Ch. VIII, Section 31) gives

CAE,

I

SPECIAL SERIES (COMPLEX

the complement of A,. Since E, * o is also carried by A,, the only possibility is that s t * o = v, * o a ~ Qd, * o = 0. Lemma 16 is thus proved. Note that llvtll < llptll < A. Proof of Proposition 2: The implication (b) (a) has already been proved in Section 8.3. To get (a) (b), we apply Lemma 16: for each t E A,, there exists an atomic measure v, carried by A,, whose norm does not exceed A and such that 4,(1) = exp 2ni1t for each 1in A. Regarding A, as a dense subgroup of R, v, becomes an atomic measure 1,carried by A' and such that 1,(x) = exp 2nixt for each x in 10, 1[ n A,. Since [O, 1[ n A, is dense in [0, 11, we have &(x) = exp 2nixt for each x in [O, 11. Let p : R + C be any bounded function whose spectrum lies in [0, I]. If we can write F(t) = jRF(x) dAt (x) = F(x) Rf{x}, there is no more = sup JFI to prove. Since l\itll= Ex,,. 11,(x}I < A, we have sup

-

-

xXEAr

< A sup Iql. The following lemma concludes the proof:

R

*P

METHODS)

167

given complex-valued continuous g : Z, -+ C will be given by a convolution f = g * p (5.4) with a discrete Radon measure p carried by A , and satisfying

the Fourier transform v of p, in the sense of distributions, v = 3 , ~is a discrete measure carried by A(&)

(5.6)

and, if I is the characteristic function of Z, and 6(t) is the unit mass concentrated in 0 Z(t) dp (t) = 6(t). (5.7) Next, by a suitable partition of unity on K, we reduce the problem to the first case.

A'

8.7. Details of the proof: con,structionof p

LEMMA 17: Let Z be a compact interval of real numbers, ,u a complex bounded Radon measure on R whose Fourier transform vanishes on I, and g, a bounded complex continuousfunction on R whose spectrum lies in I. Then jR q(x) d , ( 4 = 0. In our context ,u is the difference between Rt and the unit mass concentrated in t. A dilation and a translation reduce the problem to the case where I = [-I, 11. We define yn, n 2 1, by pn(x) = p ((1 - lln) x); the spectrum of ynlies in [- 1 + lln, 1 - lln] a r d Proposition 4, Section 12, Chapter IV, gives JR FAX)dp (x) = 0. But IvnI < 11v11 and vn(x) + p(x) as n tends to infinity. Lebesgue's dominated convergence theorem now gives JR F(x) dp (x) = 0. 8.6. Proof of part (b) of Theorem IX To start with, we show why the second part of (b) is a consequence of the first part. If lKl < 1 the union K' of K and of a point x in Q,, x E K, has the same property as K; hence the function which is 1 on x and 0 on K is the restriction to K' of a function in V Awhich vanishes on K without being identically 0. First of all, we shall prove (b) if K = Z, when A is replaced by the set A(&)of all x = mpnin Q, such that 0 < x < 1 + E. The extension f of the

Let a be a Cm real function on R carried by [0, 1 + E] and such that = 1identically. The Fourier transform a of d has the following properties: d belongs to the class Y of Schwartz, d(0) = 1 while d(j) = 0 for all j E Z, j # 0. We put b(x) = d(-x). Let p be the atomic measure carried by A, which assigns the mass b(t) to each point t in A,. As in Proposition 2, the Fourier-transform v of p , in the sense of distributions, is also an atomic measure, carried by A, n [0, 1 E]which assigns the mass a(t) to each point t of A,. Since Z, n A, = Z , the properties of b restricted to Z at once give (5.7). The estimates (5.5) follow from the following two observations: (a) (x + Z,) n A, is a translate of Z ; (b) owing to the rapid decay of b at infinity, sup Cj'=?, Jb( j + t)l ~ E R < +a. From (5.7) it is obvious that f = g on Z,.

xjEza (x -j )

+

8.8. Details of the proof: the partition of K Since ]Kl < 1 there there exist two integers m and n 2 1 such that K is m

contained in the union mp-" < 1.

U Bj of m open balls of measurep-" and suchthat 1

m

From now on we replace K by

U B,.

For each j, there is a function

1

~ ( x =) exp (2nip-"x)

- a,, which, restricted to B,, is 0 and which re-

stricted to each Bk, k # j, is a constant different from zero since each Bj is a translate of pnZp. Let 0 < E < 1 - mp-". For each continuous g : K + C, let gj be the restriction of g to Bj, and& the extension of gj to an almost periodic function whose spectrum lies in A, n [O, p-" + E];fj is given in terms of gj by a simple modification of (5.4), where Z, is replaced by B, = pnZp+ xj. ,, where dj is a constant such that on B,, We put f = dj& 1 = dj n k + j ~ l k . Then f equals& on each B,, f is almost-periodic and the spectrum off lies in A, n [O,p-" + E + (m - 1)p-"1 c A, n [0, l[. Note that the extension g -+ f from V(K) into VA is given by a continuous linear operator.

x'= nk,

8.9. How to discretize distributions on Q,? The extension operator defined in Section 8.6 can be used to atomize distributions taken in certain spaces that we now define. The notation is that of Sections 8.6 and 8.7. Let w : [0, + co[ + [I, + co[ be an increasing function and let LE(Q,) be the Banach space of all classes of measurable functions f : Q, + C such that for a constant C > 0, I f(x)l < Cw (1x1,) almost everywhere with respect to the Haar measure on Q,. We define Y(Q,) to be the vector space of all test functions g, : Q, + C which are compactly supported and locally constant. The Fourier-transform of such a g, also has these two properties. For any f E Lz(Q,), the Fourier transform S = ;Ff off, in the sense of distributions, is defined by

S(x) dx (b) for each t in A,, the mass ofL(S) concentrated in t is a(t) where a is defined as in Section 8.7; (c) the Fourier transforms of S and of L(S) coincide on Z,; (d) for a constant C > 0. (IL(S)II < CIISII, where the norms are computed in E, . In fact we put S = f E Lz, f = ( I f ) * p and L(S) = The terminology is that of Section 8.6 and the easy verifications of (a), (b), (c) and (d) are left to the reader. For each k 2 1 and each Radon measure p on Q,, let p k be the Radon measure p (pkx); if p is atomic and if the mass of p concentrated in x E Q, is a(x), p (pkx) is also atomic and the mass of concentrated ,ukin P - ~ X is a(x). Let I, be the characteristic function ~ f p - ~ Z , . A sequence L,, k 2 1, of linear atomizing operators can be found such that, for each S in Em,Lk(S) = S, is a nice sequence of approximations of S ; each S, is carried by the harmonious subset pkA, of Q, and this explains the role played by harmonious subsets of Q, in atomizing processes. If S E E,, S = where f E L,"(Q,). We define Skto be where

sf,

sf,

sf.

sfkk,

Then (5.5)-(5.7) imply that (a) S, is a measure carried by pkA,; (b) for each t in A,, the mass of L,(S) concentrated in pkt is a (pkt) jp.ct+zp, S(X)dx; (c) the Fourier-transform of S and of S, coincide on P-~Z,; (d) for a constant C independent of S and of k Z 1, llSkll < CllSll where the norms are computed in E, = OL,"; notice that pdkZPis an increasing sequence of open subgroups of Q, whose union is Q,.

9. Mean periodic functions Let Embe this Banach space of distributions; llSll is the norm of 3 in L3QP). A linear atomizing operator L can be constructed with the following properties: (a) for each distribution S in Em, L(S) is a measure carried by A, = A, n [O, 1 + E], where E > 0 isJixed from now on;

The theory of mean periodic functions provides a remarkable interpretation of coherent sets of frequencies: A is a coherent set of frequencies if and only if each mean periodic function whose spectrum lies in A is bounded or, equivalently, if and only if each mean periodic function whose spectrum lies in A is almost periodic. The exposition follows that of [47].

9.1. Engineers using Laplace transforms to solve linear differential equations of the type T * f = 0 where T is a derivation operator have noticed that this Laplace transform, originally defined in some right half plane, has a meromorphic extension F to the complex plane. Knowledge of the set (z,),, of poles of F and of the set P, {l/(z - 2,)) of principal parts of F at once gives the solution f as a finite sum Pk(t) exp z,t. Using the same idea, we study convolution equations of the type p * f = 0, where p is a given complex valued Radon measure carried by a compact interval [a, b] of the line and f is an unknown complex valued continuous function on the line. A solution of this equation is called a mean periodic function. The main theorem is the following: to get all solutions of p * f = 0 it is sufficient to consider all solutions f of the type P(t) exp At, where P is a polynomial and A a complex number, all finite sums of these particular solutions and all limits of such finite sums for the topology of uniform convergence on compact subsets of the line. Similar results can be found in 11601.

,

I,,

9.2. The space V(R)

From now on, a function will be a complex valued continuous function on the line, without any growth condition at infinity; V(R) is the linear space of all such functions. The topology of V(R) is that of uniform convergence on compact subsets of R. A measure will be a complex valued Radon measure carried by a compact subset of the line. The linear space of such measures will be denoted by ?JJlo(R).Then ?JJl,(R) is the dual space of Q(R). A mean periodic function f is defined by one of the two equivalent properties : (a) f belongs to Q(R) and the vector space spanned by all translates off is not dense in the whole of V(R); (b) f is a solution of a convolution equation p * f = 0, where p belongs to ?JJl,(R). From our point of view it is more exciting to start with such a convolution equation and to regard f as an unknown function.

9.3. The Laplace transform off Following the engineers, we solve p * f = 0 by taking the Laplace transform off. We shall prove that this Laplace transform has a meromorphic extension F to the complex plane. The determination of the poles of F and of the principal parts of F at these poles will give the approximation Pk(t) exp zkt. off by sums In defining the Laplace transform off, the first idea is to consider the integral J," f(t) exp -zt dt. This is not satisfactory since a mean periodic function can have a very rapid growth at infinity (Section 9.14). Furthermore this integral does not furnish the meromorphic extension. Our definition o f F is more unnatural: let f +(t) = f(t) if t 3 0, f += 0 if t < 0, f -(t) = f(t) if t < 0 and f -(t) = 0 if t 0. Consider the element g of V(R) defined by g = f * ,u = -f - * p. The first definition of g shows that g is supported by [a, +a[and the second one that g is supported by ] - co,b] when f is supported by [a, b]. Hence we can define the Laplace transform of g, G(z) = j,b g(t) exp (-zt) dt; G is an entire function of exponential type. Let M(z) = S,b exp (-zt) dp (t).

XkyO

+

DEFINITION 4: The Laplace transform of the mean periodic function f is the meromorphic function F(z) = G(z)/M(z). To prove the consistency of this definition, we show that F is independent of the choice of p such that f * ,u = 0. Iff * p' = 0, p' E ?JJlo(R),we define g' = f * p' = -f- * p' and we have g' * p = g * p'; hence G'(z) M(z) = G(z) M1(z)and F(z) = F1(z). Iff has a Laplace transform in the ordinary sense in g e z > x,, this f ) equals F on 9Zez > x, . Laplace transform 9( In fact +

G(z)

=

S:

=

J a exp

g(t) exp - tz dt

-zsdp(s)

=

S.

g(t) exp - tz dt

J o ft(u)exp

-zudu

9.4. Our program is to determine f by means of the set of poles of F. We need the following theorem:

X: If the Laplace transform F of the mean periodic function f THEOREM is an entire function in the complex plane, then f = F = 0. Before proving this theorem, we note the following: an entire function can be the Laplace transform of a complex valued continuous function on the line, but not of a mean periodic function (mean periodic functions are very special continuous functions on the line). 9.5. The proof of Theorem X depends on a series of easy lemmas and on a deeper proposition. LEMMA18 : Let f be a mean periodic function and o an element of n o @ ) . Then the Laplace transform of the mean periodic function f * o and the product of the Laplace transform o f f with jRexp -zt do ( t ) difer by an entire function. Proof: We have f+ * o = ( f * o)+ + r, where r is a compactly supported bounded function. Assume that f * p = 0 and put g = f + * p, h = ( f * o)+ + p. Then g * o = h r * p, which gives G(z)S(z) = H(z) + R(z) M(z), where R(z) = jRexp ( - z t ) r(t) dt is an entire function. The Laplace transform off is G(z)/M(z)and that off * o is G(z) S(z)/M(z) - R(z), which completes the proof.

+

LEMMA19: Let p be a complex valued Radon measure carried by [a,b] and A a complex number such that j : exp -It dp ( t ) = 0. Then there is a complex valued function m with bounded variation on [a, b] which is 0 outside [a, b] and such that, for each complex z f 1,

J

a

exp (- zt)m ( t ) dt = -J e x p - ~ t d p ( t ) . z-1 0

Proof: Replacing dp ( t ) by exp (-1t)dp (t),m(t) by exp (-1t)m ( t ) and z - 1 by Z, we can assume that 1 = 0. We simply take m(s) = y-, dp (t). From now on we shall write p~c,instead of m(t) dt. LEMMA 20: With the hypothesis of Lemma 19, let f be a mean periodic function, F the Lapluce transform o f f and1 a complex number which is not a pole of F. Then f * p = 0 implies f * p, = 0. Proof: As in the proof of Lemma 19, we may restrict our attention to the case 1 = 0. Then using the notation of Lemma 19,f + m has a distri-

butional derivative equal to f * p = 0. Hence the continuous function f * m is equal to a constant a. If a = 0, the lemma is proved; if a f 0 , the Laplace transform off * m is alz Lemma 18 shows that 0 is a pole of F, which by hypothesis is not the case. 3 : Let p be an N-times dzTerentiable function carried by PROPOSITION be the sequence of all zeros of the entire function [a, b],N 2 4. Let (Ak)kaO M(z) = j!: exp ( - z t ) p ( t ) dt. There exist an integer n 2 0 and a complex constant c such that

where p, is dejined as in Lemma 19 and where the series is uniformly convergent on [a,b]. Proof: The idea is to take Fourier transforms of the two sides of (5.9) and to establish convergence in L1 (- co, + co) for these Fourier-transforms. Returning to the original functions, we obtain uniform convergence. First of all, let us write down the Laplace transform of each side of (5.9). We obtain

But (5.10) is a consequence of Hadamard's factorization of entire functions of exponential type: ~ ( z=) C e c Z z n n ?(1 - ~ 1 1 ~ taking ) ; the logarithmic derivatives we get (5.10) with uniform convergence on compact subsets of C. In proving convergence in L1 (- co, + oo) of the restrictions to the imaginary axis, the following lemma will be useful: LEMMA 20 : Let

We have for a constant C and all real y, ly( 2 1 ,

The sequence of zeros of an entire function of exponential type satisfies

1: 11k1-2< + oo ([16],th. 8.2, p. 328) and we have N > 4 (the value o f N is given by the regularity of p). Hence Lemma 20 gives Proposition 3. To prove Lemma 20 we have to consider three cases:

(a) lz - 1,1 > litk1/2implies IXk(z)l < 2/21 IM(z)l 14k(-2by the second form of X,; (b) llk1/2 2 lz - Akl > 1 implies IXk(z)l < 21zI2 IM(z)l lAkl-2, since bl 2 I&1/2; (c) if lz - A,[ < 1 and lil,l 3 2, then (M(z)/(z - ilk)( < sup 12'-11=

2

IM(z')/(z' - A,)], since M(z)/(z - A,) is an entire functicn. Hence (M(z')I, where we have here lz' - A,[ 3 1 SUP IX,(z)[ < 2 12'-zl= 2

and lz/Akl < 2 Iz/Ak12by the assumptions on z and 1,.But from a direct evaluation of M(z) = J,b exp -ztp (t) dt, we obtain IM(z)l < ~ e ~ ' l " l / ( z ( ~ , where z = x + iy. Substituting this estimate in the inequalities obtained in (a), (b), (c) we get (5.11) with perhaps another constant C. Note, finally, that if ]jlkl < 2 and lz - a,/ < 11,1/2 we get lzl < 3 and there is nothing to prove. 9.6. Proof of Theorem XI Assume that the Laplace transform off is an entire function and that f * p = 0; convolving this equation with a regular compactly supported function, it may be assumed that p is N-times continuously differentiable. Let (A,),,, be the sequence of zeros of M(z) = JR exp (-zt) p(t) dt. , 0 for all k 2 0. Lemma 20 shows that f + p ~ = By the uniform convergence of (5.9), we get f * (tp) = 0. Iterating this, we obtain f * (Pp) = 0 for all polynomials P. Let to be a real number such that p(to) # 0. Then the Dirac mass E,, concentrated in to is the limit of a sequence P,p in the weak-star topology a (%,(R), %(R)). Passing to the limit, we get f * = 0 and f = 0. 9.7. THEOREM XI: Let p be a complex valued Radon measure carried by a compact set of real numbers. The set of all solutions off * p = 0, where f is a complex valued continuousfunction on the line, is given by the following set of rules: (a) all solutions h of h * p = 0 of the type h(t) = P(t) exp At, where P is a polynomial and 1a complex number must be found; Pk(t)exp 1,t of solutionsfound in (a) must be (b) allfinite sums g = written, (c) all limits f of convergent sequencesgj, j 1 of solutionsfound in (b) must be taken where the topology is that of uniform convergence on compact subsets of the line.

I,,

Before giving the proof, a remark should be made. The degree m of P and the complex number 1concerned in (a) can be found as follows: 1is a zero of the entire function M(z) = jg exp -zt dp (t) and m is the multiplicity of this zero.

9.8. Now to the proof. Let E be the space of all functions g defined in (b); E and the closure E of Eare translation invariant; B i s determined by the following rule : the function h belongs to E if, for each v E %,(R) whose Laplace transform N(z) at least vanishes whenever the Laplace transform M(z) of p vanishes, the convolution h * v is identically 0. We have to prove that f belongs to $ and to this end we show that k = f * v is identically 0. This is the case if the Laplace transform K(z) of k is an entire function (Theorem X). By Lemma 18, K(z) - F(z) N(z) is an entire function. But, by definition (Section 9.3) of F(z), F(z) M(z) is an entire function. Since N(z) vanishes at least whenever M(z) vanishes, N(z)/M(z) is also entire. Hence K(z) is entire, k = 0 and f E l?.

9.9. The spectrum and the Fourier series of a mean periodic function Let f be a mean periodic function, F the Laplace transform off, (zj),> the sequence of poles of F, Pj (l/(z - z,)) the principal part of the expansion of F in a neighbourhood of zj and mj the multiplicity of the pole zj or the degree of P,. DEFINITION 4: With the above notation, putting zj = 2niAj, we define the spectrum off to be set of all pairs (4, mi), j > 1. DEFINITION 5: The Fourier series off is the formal series exp 2ni1,t. The argument used in Section 9.8 gives the following result:

Pj(t)

XII: For the topology of uniform convergence on compact THEOREM subsets of R, each mean periodic function is a limit of jinite sums fk(t) = PjSk(t)exp 2ni;lJt. The degree of P,,, equals the degree mj of P,, and the coefficients of Pi,, tend to those of Pj as k -+ + co. To prove this last remark, the following lemma will be needed:

zja

LEMMA 21 : Let A be the spectrum of a mean periodic function f and (fi),,, a sequence of mean periodic functions whose spectra lie in A and

9.11. It remains to study sets of zeros of entire functions of exponential type having a slow growth on the real axis. For a precise statement of Beurling-Malliavin's theorem, a new definition will be needed.

which converges to f uniformly on each compact set of real numbers. Then the Laplace transforms Fj offj tend to the Laplace transform F o f f in the following sense: on each compact subset K of complex numbers not containing a pole of F, Fj tends uniformly to F and if 1 is a pole of F of order m , then 1 is apole of F j of order < m and the coefficients of the Taylor expansion of F j in a neighbourhood of 1 tend to those of F. Proof: Let p be an element of '2R0(R) such that f * p = 0 and let M(z) = 1exp (- zt) dp (t). Let g = f * p, gj = f: * p, G(z) = j exp (- z t ) g(t) dt, Gj(z)= j exp ( - z t ) g j ( t ) dt. Thenfj * p = 0 since the spectrum of f, is contained in that off. The Laplace transform off is G(z)/M(z)and that off, is Gj(z)/M(z).The supports of gj are contained in a fixed interval [a, b] and gj tends to g uniformly on [a, b]. Hence Gj(z)-, G(z) uniformly on compact subsets of C, and lemma 21 is proved.

,

+

x,>

9.10. How to construct mean periodic functions? Let A be a sequence of pairs (A,, m k ) k a where A, E C and m k E N. Is it possible to construct a mean periodic function whose spectrum is A? Beurling and Malliavin [30]give a complete answer to this question. We shall state the results. The problem can be divided into two parts.

XIII: There is a mean periodic function whose spectrum is A if THEOREM and only if there is an entire non zero function A(z) defined in the complex plane and satisfying A(A) = 0 for each 1 E Aand IA(x)l < C ( 1 1 ~ 1 for ) ~ suitable C > 0 and N 0 ( x E R)

+

IA(z)l < C exp llzl for suitable C > 0 and I > 0 ( z E C ) . Proof: The condition is necessary. I f f is mean periodic there is a con?plex measure p carried by some compact interval such that p * f = 0. By definition of the spectrum A(z) = jR exp (-2nizt) dp ( t ) vanishes on A. We define the complex Fourier transform of a compactly supported distribution S on the line to be A(z) = JR exp (-2nizt) S ( t ) dt. The condition is sufficient. By Paley-Wiener's theorem, A is the complex Fourier-transform of a distribution S carried by [-I, I]. Convolving S with a suitable test function pl in 9 , we may assume that S is an element of 9 , not equal to 0, whose complex Fourier transform vanishes on A. It remains to construct f as a series 2: aktmkexp 2niAkt ( A = (A,, mk),>J, where a, > 0 tends to zero rapidly enough to ensure uniform convergence on compact subsets of R. Hence f * S = 0 and f is a mean periodic function.

6: The mean period associatedwith a setA ofpairs(A,, m,),, DEFINITION (A, E C , mk 2 0 ) is the infimum p(A) of lengths of intervals [a, b] such that there exists a non-zero complex Radon measure ,u carried by [a, b] whose complex Fourier-transform1 vanishes on A. Remark 1 : I f [a, b] is an interval whose length is less than p(A), each complex valued continuous function defined on [a, b] is the limit of a sePk(t)exp 2niAkt, where dOPk< mk quence of finite sums Proof: Apply the Hahn-Banach theorem. Remark 2 : If [a, b] is an interval whose length is greater than p(A), each mean periodic function f whose spectrum lies in A and which vanishes on [a, b] vanishes on the whole real line. Proof: Let p be a measure carried by [a', b'], b' - a' < b - a, such that the complex Fourier-transform of p vanishes on A. The problem is clearly translation invariant and so we may assume that a = 0 Let f be defined by f + ( t ) = f ( t ) if t >, 0 and f + ( t )= 0 if t < 0. Sincef vanishes on [0, b] and since p is carried by [a', b'] whose length is less than b, f * p = f * ,u = 0. Using the definition of the Laplace transform F off, we get F = 0. Hence f = 0 by Theorem I. As a corollary we have that any mean periodic function whose spectrum lies in A is fully determined by its values on a compact subset depending only on A. It is interesting to compute the length of such an interval of defin~tion.Beurling and Malliavin's theorem gives the exact value of this length in terms of the exterior density of A, which we shall now define.

I

+

+

9.1 2. A density due to Beurling and Malliavin Let Q be a set of open intervals on the line (the notation o E 0 denotes intervals of this set), and let T(Q) be the union of all squares T ( w ) of R 2 one of whose diagonals is on w E Q : if x is the middle point of cu and 21 the J X - xl < 1. length of o, T ( o ) is defined by I YI A set 0 of intervals is said to be negligable if ( 1 + x2 + y2)-I dx dy < + co. (Note that for any set of open intervals, T(Q) is an open set of squares and is a measurable set.)

+

The complex Fourier-transform of ,u is 9: C -t C defined by P(z) = J: exp - 2nizt d~

w.

For example, if there is an I > 0 such that the lengths of the o E Dare all less than I, 9 is negligable. Roughly speaking, 2 ! is negligable means that it is improbable that 9 contains arbitrarily large intervals. A complex valued (not necessarily bounded) Radon measure p on the real line is said to be regular and has the density a if, for each positive E, the set 9,of all open intervals o of length lo1 such that llwl-l jm dp - a1 2 E is a negligable set of intervals. We then write a = A (dp). To each setA of pairs (A,, mk)kzl of real numbers together with multiplicities, we attach a positive measure dNAdefined by m, 6 (x - A,), where 6(x) is the unit mass concentrated in 0 and we assume that k -+ A, is an injective map.

C,,,

7 : The exterior density ofA, d,(A) is inf ( A (dv)) over regular DEFINITION positive measures dv such that dv 2 dNA. THEOREM XIV: With the above notation, (1) The set A of pairs (A,, m,),, is the set of frequencies of a mean periodic function f(t) on the line if and only ifd,(A) isfinite; (2) The mean period associated with A is d,(A). The proof of this result is deep and difficult. Since this theorem will not be used in the sequel, the reader is referred to [30]. An easy corollary can be given. We recall that a set A of real numbers is called regular if there exists a d > 0 such that each interval of length d contains at most one point o f A and a D > 0 such that each interval of length D contains at least one point ofA.

,

4: For a regular A, the mean period p(A) of A lies in the COROLLARY interual [ l / D , 1Id]. Proof: Consider R as the union of all intervals [kd, (k + 1) d[, k E Z and construct a measure dv as follows: if [kd, (k + 1) d[ n A = {A}, unit mass is concentrated in L and if [kd, (k + 1) d[ does not intersect A, unit mass is concentrated in kd. Hence dv 2-dNAand dv is regular and has the density lld. We thus obtain p(A) < lld. To get p(A) 2 1ID, we merely apply Theorem 111. 9.13. Mean periodic functions whose spectra are contained in a given A

From now on, A is assumed to be uniformly discrete, i.e. there exists a d > 0 such that each interval of length d contains at most one point ofA. We assume also that A does not present multiplicity (all m, = 0).

is then the Frechet space of all mean periodic functions whose spectra lie in A ; the topology V is that of uniform convergence on compact sets of real numbers. It is remarkable that the vector space of all mean periodic functions is not complete in the topology V ; the vector space of all mean periodic functions whose spectra are contained in a given A for whichp(A) < co is complete for %? since there exists a fixed measure ,u with compact support such that ,u * f = 0 for all such$

+

9.14. Behaviour at infinity

All f in V Aare fully determined by their values on any interval I whose 1 1exceeds p(A). The fundamental problem is now the following: length 1 is there a method of computing or estimating the values off outside I in terms of the values taken by f on I ? The answer is given by the following theorem:

+

XV: Let o : R -, [l, co [ be a function andA a regular set of THEOREM real numbers: each interval of length d contains at most one point ofA and each interval of length D contains at least one point of A. The following assertions are equivalent: (a) each mean periodic function f whose spectrum lies in A is 0 (o(t)) a s It( tends to in$nity; (b) there exist a constant C and a compact set K of real numbers such that, for each mean periodic function f whose spectrum lies in A and for all real t, If @)I < Cw (t) sup If I. K

Proof: Let ~?8be the Banach space of all complex valued continuous functions which are 0 ( ~ ( t ) as ) It( -t co.The norm of such an f in is sup Iw-lf 1. Let I: V A-+ LA? be the injection defined by (a). The topology

+

R

of W A is defined by the fundamental sequence of neighbourhoods of 0, V , = { f E V A: sup If 1 < 2-,}. The closed graph theorem shows that I C-f,kl

is continuous. Hence the inverse image under I of the unit ball of 9 contains a V , for sufficiently large k and this gives (b). The converse implication is trivial. Example: Let 8 > 2 be a real number and A, the set of all finite sums &,Ok,E , = 0 or 1. If 8 is a Pisot number, all mean periodic functions whose spectra lie in A, are almost periodic functions as we shall prove later. If 8 is not a Pisot number, mean periodic functions whose spectra lie in A, are known as soon as they are known on an arbitrarily small

interval [0, I] but it is impossible to give estimates for these functions in terms of their values on a compact set of real numbers. To substantiate these two remarks, two observations can be made: (a) if 8 > 2 each interval [x - T, x + TI of real numbers contains o(T) elements ofA and the little o is uniform with respect to x ; (b) there exists1 a sequence P,(t) of finite trigonometric sums whose frequencies belong to A such that P,(O) = sup IPk(t)l = 1 while P, tends R

uniformly to 0 on each compact set of real numbers not containing 0. Assertion (b) in Theorem XV cannot be true, since for s 4 K, t -+ P, (t - s) converges uniformly to 0 on K and takes the value 1 at s. THEOREM XVI: Let A be a regular set of real numbers. The following three assertions are equivalent: (a) each mean periodic function whose spectrum lies in A is bounded; (b) each mean periodic function whose spectrum lies in A is almost periodic; (c) A is a coherent set of frequencies. Proof: (a) * (c) by Theorem XV. Since trigonometric sums are dense in VA for the topology of uniform convergence on compact sets of real numbers (Theorem XII) and since by (c) this topology is equivalent to that of uniform convergence on the whole real line, each mean periodic function in VA is almost periodic. Hence (c) * (b). It is obvious that tb) =, (a).

Then A is the Fourier transform of a positive function b which is 0 ( r 2 ) as Proof of: rT

Let I > 0 be such that inf b(t)

> B > 0 (if E 6 1/4d, such an lexists).

c-1,ll

Then Parseval's equality gives l/b~11;= I&eA aAA(x - A)I2 dx = ll All: I r o n lan12.Hence 2 jk1 IP(t)I2 dt 6 11~112 laAI2.Since the space of trigonometric sums is translation-invariant, we get, for each k E Z, e2 jI::-f::: IP(t)12 dt 6 ~ ~ A I I : laAI2and adding a sufficiently large number of these relations we get (5.12). Proof of:

xleA

xAeA

c1AeA Z Ian126

-T

IP(t)12 dt.

:::

Eke,

(5.13)

2

We start as before with I ~ ~$ P =I I j$i+ b2(t) IP(t)I2 dt = /[A11 la,12. Since b(t) is ~(t-') as it1 + +a, 11b~11226 jTTb2(t) 1p(t)l2dt T-lC, @ ' , lai12 (using 5.12). Since b(t) 6 1/2d, we get I?, IP(t)12dt > 4d2 (llA112 - T I C 3 ) lanI2.If Tis large enough, (5.13) is proved. Notice that the L2 theory is easy and has a metric form while the L" theory (equivalence between sup If 1 and sup If I) is deep and is connected x A E A

+

rneA C-T,TI

R

with the arithmetical structure ofA.

9.15. The L2 theory We say that a set A of real numbers is uniformly discrete if there exists a d > 0 such that the distance between any two distinct points inA exceeds d. THEOREM XVII : There exist a T > 0 and two constants C1 and C2 > 0 depending only on d > 0 such that, for each mean periodic function f whose spectrum is simple and lies in A, we have rT

znoA

where the Fourier series off is an,exp 2nih. By Theorem XII, we can restrict our attention to finite trigonometric sums P(t) = an exp 2niilt. Let A :R + [0, + co[ be defined by A(x) = sup (0, 1 - 21x]/d).

xi,,

See Lemma 6, Section 5 of Chapter IV.

10. Notes IfA is a set of real numbers and if E is a Lebesgue-measurable subset of the line, we write A 4 (E, 2) if there exists a constant C such that for each trigonometric sum P(t) = A aAexp 2niAt, lanlZ< C j, (PI2dt 6 C2 laAI2. For the case where E is an interval, this problem has been widely studied and the infimum of the lengths of those E for which A 4 ( ~2), can be computed [48]. If E is a finite union of intervals, the problem becomes quite difficult: A 4 (E, 2) implies (El > &Zi (A), where I 1 denotes Lebesgue measure and the definition of the upper density ofA and the proof are in 1541. The converse result is plainly false, even if A = Z. The L2 version of Theorem VII gave the Erst example for which IE ( > dens (A) implied 4 ( ~ 2). ,

C

CAE

182

SPECIAL SERIES (COMPLEX METHODS)

It would be interesting to give other examples for which the latter implication is true and where the arithmetical structure ofA is not so explicit. For lacunary sequenced, some partial results have been obtained. For each Sidon set A and each Lebesgue measurable E of positive measure, A 4 (E, 2) holds 1311, although this hypothesis on A is not necessary for this conclusion [32]. An n-dimensional version of Theorem XI is given in [60].

CHAPTER VI

SPECIAL T R I G O N O M E T R I C SERIES (GROUP-THEORETIC METHODS)

T l s chapter is divided into two parts. The first one is devoted to topological Sidon sets A of real numbers: each bounded function b :A -* C is the restriction toA of the Fourier-transform of a complex bounded Radon measure p on R. For almost periodic functions f whose spectra lie in A, there is a diffusion of regularity: iff is continuously differentiable on an arbitrary small interval, the derivative off exists on the whole line and is almost periodic. Sidon topological sets are stable coherent sets of frequencies; in general, they are not harmonious sets. In the second part, we show that if (A,),,, is an increasing sequence of real numbers such that A, # k for infinitely many k > 1 and A, - k -, 0 as k -+ + co,the setAofA,, k > 1, never has this property. A continuously differentiable almost periodic function f whose spectrum lies in A and whose derivative is not bounded can exist; A is not a coherent set of frequencies. The proof of the last result uses the determination of idempotent or semi-idempotent measures on a compact group.

A. Topological Sidon sets of real numbers 1. Definition and basic properties of topological Sidon sets DEFINITION 1 : A set A of real numbers is a topological Sidon set if each complex-valued bounded function dejined on A is the restriction to A of the Fourier transform of a complex valued bounded Radon measure p on R. THEOREM I: L e t A be a set of real numbers. Thefollowing two statements are equivalent: (a) A is a topological Sidon set; (b) there exist a compact set K of real numbers and a positive constant C such that, for all trigonometric sums P ( t ) = cl exp 2niAt, we have Iden Ic~l < C SUP IPI.

Ed,,

K

Proof: (a)

(b): The hypothesis on A yields a linear map T : l m ( A ) the closed graph theorem shows that T is continuous. Hence there is a constant C , > 0 such that for each bounded function b : A + C, we can find a measure p in M ( R ) satisfying the two conditions llpll < C1ll bll, and P(1) = b(1) for all il in A. Theorem VI of Chapter IV shows that A is a coherent set of frequencies. Hence there is a compact set K of real numbers and a positive constant C , such that, for each function b in B(A), we can find a measure p carried by K such that llpll Q C211bllBcA, and P(1) = b(1) for all 1 €A. Combining these two facts, we ficd a compact set K of real numbers and a positive constant C such that, for each bounded function b : A --+ C there is a measure p carried by K whose norm does not exceed C sup Ibl A and whose Fourier transform restricted to A is b. Now let P(t) be a trigonometric sum '&A c, exp 2ni1t. We define b : A -t C by setting b(1) = 1 if c, = 0 and b(1) = c,/lc,l otherwise, so that ]b(1)]= 1. Let p be a measure carried by K whose norm is less than C and whose Fourier-transform restricted to Ais b. Then jK P(t) dp ( t ) = cA P ( 4 = ~ Z E AIcA Hence xnsn IcAl G C SUP IPI.

2. Examples of topological Sidon sets

-, B(A) and

InEA

K

(b) (a) : Let b :A -+ C be any bounded function. Let d be the normed space of the restrictions to K of all trigonometric sums ElenC, exp 2 n i h with the norm sup IPI. Then b defines a continuous linear form L on K

8 by L(P) =

XA,A -c,$ (4;from (b) we get IL(P)l < CnEnlanl Ib(L)I < C ll b ll ,

sup (PI. By the metric form of the Hahn-Banach theorem, L can be exK

tended to a continuous linear form on the whole of C(K). Hence we get a complex valued Radon measure p such that llpll < Cllblla, and @(A) = b(A) for all 1 in A. Remark: A Sidon set A in R can be defined by the following property: there is a constant C such that for each trigonometric sum P(t) = CnoA cA exp 2ni1t we have Ic,l Q C sup I PI. A topological Sidon set A is then

In,

R

defined by the two conditions: A is a Sidon set and A is a coherent set of frequencies. We use the qualification 'topological' to distinguish 'topological Sidon sets7from Sidon sets, for example, the set of all 2-,, k 3 0, is a Sidon set but is not a topological Sidon set.

PROPOSITION 1 : Let (A,),, be an increasing sequence ofpositive real numbers such that A,+ 3 41, for all k B 1. Then the set A of all A,, k B 1 , is a topological Sidon set. The proof of this proposition depends on the following lemma:

,

,

LEMMA 1 : Let (b,), be any sequence of complex numbers of absolute value 1 and (A,),, an increasing sequence ofpositive real numbers such that A,,, > 41,. Then there is a real number s such that Is1 < 1/1, and lexp 2ni1,s - bkl < 1 for each k > 1. Proof: The inequality lexp 2ni1,s - b,l < 1 is satisfied when s belongs to the union of closed intervals I,,, of lengths 1/31, whose centers belong to a suitable coset of 2, '2. Since A,,, B 41, it is very easy to see that Hence each I,,, contains each I,,, contains a complete interval such a solution s. Proof of Proposition 1: Let P(t) be any trigonometric sum C, exp 2ni1,t. We define b, by bkck = lckl if ck # 0 and by 6, = 1 if c, = 0. Then P(s) = z,, where lzkl = 1 and lz, - 11 Q 1 which imply We z, > 3. Hence g e P(s) 2 &, I lckl and x k r 1 lckl Q 2 sup IP(t)l. (tlS2l-i It will be proved later that any finite unionA of topological Sidon sets is again a topological Sidon set whenever there is a positive d such that d, 1 EA,1' €A.Applying this result in our situation, we get inf 11 - 2'1 the following theorem:

Ckrl

xk,ll~kl

,

11: Let ol be a real number greater than 1 and let (A,),, be an THEOREM increasing sequence of positive real numbers such that, for all k 2 1 , A,,, > nil,. Then the set A of all A,, k 2 1 , is topological Sidon set. 3. Construction of remarkable measures associated with topological Sidon sets 3.1. If A is a topological Sidon set, every bounded function b: A -, C is the restriction to A of an element 6 of B(R) but this statement does not give any information about the values of 6 outside A. The following theorem shows that we can be far more precise about 8. THEOREM 111: L e t A be a topological Sidon set of real numbers. There is a constant C > 0 such thatfor eachpositive E , eachpositiveq, andevery bound-

ed function b :A + C, we can find a measure p with the followingproperties: Ilpll < C5-l SUP Ibl (6.1) A

$(A) I$(t)l

< E sup Ibl A

=

b(1) for all 1 EA

if the distance from t to A exceeds 7 .

(6.2) (6.3)

Notice that the norm ll,ull of ,u does not depend on q > 0. IfA is a set of integers, then A is a topological Sidon set if and only if c,. there is a constant C such that for every trigonometric sum P(t) = exp 2ni1t, we have &, Ic,l < C sup IPI. The proof of Theorem 111then

3.3. A 'random' space Q used in the proof of Theorem III Let { - 1 , 1 ) be the multiplicative group of two elements a n d 9 = { - 1,1) the product of k copies of (- 1, 1). The Haar measure d o on 9 is defined by the condition that each point of SZ is assigned the mass 2-,. The dual group of 9 is the group G of all sequences g = ( E ,, ...,E,) of 0 or 1, where addition is defined to be componentwise addition modulo 2. Each function f : SZ + C has a Fourier series

xneA

C0.11

gives the following result. For some constant C > 0 and for each positive E , there is a measure p carried by [O,2n] such that ll,ull < CE-l. $(A) = 1 if 1 EA, and /$(j)l < E if j E Z, j $A. If the norm of ,u were independent of E > 0, ,u would tend to a weak limit v as E tended to 0. Such a measure v is carried by [O,2n],$(A) = 1 when 1 EA and 9(j) = 0 for all other integers j. The existence of v is impossible by Theorem IX below.

Parseval's relation gives

3.2. In the proof, we can confide our attention to the case where b(1) = 1 for all 3L in A. Let ,ul be the measure obtained in this case. Each bounded function b :A + C is the restriction to A of the Fourier transform fl, of a measure p, whose norm does not exceed C, llbll m. Theorem I11 is proved if we put p = p1 * ,u2. Since A is countable, it can be presented as a sequence l . AS a first step, A is replaced by A, = ( 4 , ..., 1,) and a measure p, is constructed such that (1) ,uk is carried by a fixed compact set K, independent of k ; (2) llpkll < CIS; (3) @,(A) = 1 for all 1 €Ak; (4) I$,(t)l < E if the distance from t to A, exceeds q. Let Q(K) be the Banach space of all complex valued continuous functions on K and let o (%R(K),Q(K))be the topology of pointwise convergence of measures tested on such continuous functions. We can find a subsequence of the p, which converges to a measure p satisfying conditions ( I ) , (2) and (3) of Theorem 111(with b = 1). ,ukwill now be constructed. Let ( K , , C,) be suitable for A: this means that for each finite trigonometric sum P(t) = FLEAc, exp 2ni1t we have &,A IcAl < C1 sup IPJ. Then (K, , C,) also has this property for A, and,

3.4. 'Random' measures p, used in the proof of Theorem III

K1

from now on, we write p instead of pk and A instead of A,.

Since (Kl , C,) is suitable forA,, we can find a measure v, for each o in 9 such that IIvmII < C 1 ,v, is carried by K and $,(Aj) = oj far 1 < j < k. Let ,urn be defined by p, = j, v,,. * v,. dw'. This convolution is, in fact, a finite sum. Hence p, is a measure; Ilp,ll < c:, PO (Aj) r-1 = j, ojoj oj do' = wj JQ do' = o+ and, for all real t,

where sj

=

0 or 1 and (el.

1....

Ic ( t ;E ,

, ..., &,)I

< C: .

&k)

This last inequality shows that the Fourier coefficients of the 'random' ,, are more regular than the Fourier coefficients of the 'random' measure u measures v,; the regularity is required with respect to cu E Q. To prove this last inequality, note that

By Parseval's formula =

d2 ( t ;E,,

...,E,)

(el.

C

....ek)

Id ( t ;E , , ...,&, ) I 2

which proves (6.6).

< C12 . But c ( t ;e l , ...,E,)

3.5. Riesz products R,

We now define 'random' Riesz products R,: R -+ C by R,(t) = - 1 - i + n 2 1 + smj cos Ajt) i (1 + EWJ sin Ajt). The following statements are easily seen to be true. (a) R,= -1 - i + A,+ iB,, where A,>,O, B,>O; hence JR,I d A, B, 2; (b) Sf2 IR&)l dm < 4; OJ exp iAjt s2E r r (sl , ..., sk, t) mE: (c) R,(t) = E w?, where r (sl, ,..,E ~t), is a product of two or more cosines added to i times a product of two or more sines and where the dash means that the last sum + sk 3 2. is extended over all sequences of 0's and 1's such that s1 + The Fourier-transform1 of R,(t) (with respect to t) is the measure

+

+

nt

+

The Fourier transform of R, f is given by k

(R, f ) " = f

A

* SR,

mjf(t - Aj)

=s 1

+ s2

x'

m?

m?'gE1, ...,Ek(t),

+

2:

where g,,, ...,,, - p,,, ...,,, * f and hence llg,,, ...,ekll < 2; the last sum is extended over all sequences (s,, ..., E,) of O's and 1's such that s, + ... + Ek > 2. Then (6.8) and (6.9) give

where 6(x) is the unit mass concentrated in 0 and where p,,, ...,,, is an atomic measure whose norm does not exceed 2. 3.6. A function f

The function f is 1/2n on [-n, n] and 0 outside. Then Il f 11, = f(0) = 1, and if n is chosen large enough, ~J'(t)l < s whenever It1 3 17. With such a choice of n, we put [-n, n] = K2 . 3.7. A formula for p

Sf2

We define p to be p, * (R, f ) dm. This convolution is taken in !Dl(R) and , j denotes a finite sum. 3.8. Properties of p

p is carried by K,

11~11

jn lllu~ll l &f

+ K, + K2 = K which is independent of k 11 dm

9 ,

2

C: So llRwf 11 dm = C I ~ Q X RIR,(t) f(t)l dt. Reversing the order of integration and using the inequality Jf2 IRw(t)ld o < 4, we get llpll < 4 ~ 2 . The Fourier transform o f f is f ( x ) = Jf 2 e-lxY(t)dt.

If the distance from t to A, exceeds q, we use (6.1 1). Since ...,El' lc (t; and ~g&ls...~&k(t)~ G 2, we have Ib(t)l < 2c2 ~ . f (-~ El . . zk)l < 2 ~ 2 ~ ~ If t = A,, 1 < j < k, then ,A,(t) = 0,. Using (6.10) we obtain $(t) = s. Finally p has the following properties: llpll 6 4 c f , /?(A) = s if A e A k and I,d(t)l G 2c;s2 if d (t,Ak) > q ; p is carried by K = K, + Kl + K2 which is independent of k. Replacingp by s - l p and then E by s/CI h,we obtain the requiredp,. 4. The union of two topological Sidon sets From now on let A be a uniformly discrete set of real numbers. This means that there exists a d > 0 such that for all 1and 1' in A, A # A' implies 1 1- A'( 2 d.

190

SPECIAL SERIES (GROUP-THEORETIC METHODS)

THEOREM IV: Ifrl is the unionA, u A, of two topological Sidon sets, A is again a topological Sidon set. The proof depends on the following lemma:

2 :LetA be a set of real numbers, K a compact set of real numbers LEMMA and C a positive real number, and assume that, for any function b : A -P C, there is a complex valued Radon measure p such that (a) p is carried by K; (b) Ilpll < c ; (c) lg(1) - b(1)l < 4for all A in A. Then A is a topological Sidon set and for all trigonometric sums Ic,l < 2C sup ]PI. P(t) = '&,A c, exp 2 d t , we have

ILEA

K

Proof: Define b(31)to be c,/(c,l if c, # 0 and to be 1 if cA = 0.Then, for -c,,d (1) the corresponding p given by the lemma, JK P(t) dp ( t ) = = xnGnca (b@) - b ( 4 ) + ~ A ICAI.~ Hence A g e S,. P(t) dp ( t ) 2 5 ,A I CAI. But B e j, P(t) dp ( t ) < C sup IP(, which completes the proof.

I,,,

1

K

Two uniformly discrete sets of real numbers A and A' are said to be adjacent if there is a 1-1 mapping p : A-+A' such that p(31) - 1 + 0 as

+

1311 + oo. A set of real numbers A is a stable coherent set of frequencies if each adjacent A', including A itself, is also a coherent set of frequencies. THEOREM V : Stable coherent sets of frequencies are Sidon topological sets and conversely. For any closed set E of real numbers, let B(E) be the Banach algebra of all restrictions to E of the Fourier transforms of complex bounded Radon measures p on R ; llyllBcE)= inf {IIpII ;$ I E = Q)}. The proof is based on the following lemma: LEMMA 3: Let A be a topological Sidon set. Then there exist a positive 1, (21
Proof of Theorem IV: Let b : A + C be any function such that Ib(31)l = 1 for all L E A. Let b , (resp. b,) be the restriction of b to A, (resp. A,). Using Theorem 111, we find two constants C , and C 2 ,two compact sets of real numbers K, and K, (not depending on b) and for each positive E two measures p, (depending on s), v, (depending on b,), and two further measures p2 (depending on s), v2 (depending on b,) such that (a) 1121 = 1 onA1, l,&l < 8 0 n . 4 , IIpl/I < C1e-l; (b) @, = b o n 4 , IIv, I 1 < C1; (c) 1122 = 1 onA2, l P 2 l < E o n 4 Ilp2Il < C2s-I; ( 4 92 = b o n A 2 , llv211 < C2. We then put p = p, * v, p, * v2; p is carried b y K = ( K , + K,) u (K2 + K2) and Ilpll < s-I (c: c:). On A , , @(A)- b(I)l < C2&;o n A 2 , @(A) - b(I)l < Cis. If C1&< 4and C2&< 4the hypothesis of Lemma 2 is valid and Theorem IV is proved.

Sidon set is a coherent set of frequencies. The union of a topological Sidon set with a finite set is also a topological Sidon set. To prove the stability of topological Sidon sets, it may be assumed that for each 1 EA,131 - p(1)I < 1. Let b' :A'+ C be any bounded function and let b : A + C be b' op. Then, in the notation of Lemma 3, b can be extended to a function 8 : A + [-I, I]+ C whose norm in B ( A [-I, I]) does not exceed Allbllm = AJlb'I), The restriction of 6 to A' is b' and is the restriction of an element of B(R) whose norm does not exceed AIlbtllm. Hence A' is a topological Sidon set. Conversely, let us assume that A is a stable coherent set of frequencies. We can clearly find a set A' of real numbers with the following two properties: A' is linearly independent over the field Q of rationals and A' is adjacent to A. If we can prove that such a d ' is a topological Sidon set, by the first part of the proof, A will also be a topological Sidon set.

5. Topological Sidon sets and stability of coherent sets of frequencies

LEMMA 4: IfA' is linearly independent over Q and is a coherent set of frequencies, then A' is a topological Sidon set. Proof: Each function x :Af+ T is a weak character on A'; hence, by Theorem VI of Chapter IV, x E B ( A f ) and I I x ~ ~ ~ < ~ ~ A. , ) But each function b :A' -+ C which is bounded by 1 can be written b = (x, + x2) for two such X . Then b E B ( A f )and IlbllBcn.,6 A. Lemma 3 can 5e improved.

+

+

A set of real numbers A is a coherent set of frequencies if there is a compact set K of real numbers and a positive constant C such that for every trigonometric sum P(t) = c, exp 2ni1t, sup I PI < C sup 1 PI.

I,,,

R

K

A set of real numbers A is uniformly discrete if dA = inf {II' - 311;

A E A , I ' E A , 31' # 1) is a positive real number.

.

+

+

192

SPECIAL SERIES

(GROUP-THEORETIC METHODS)

THEOREM VI: L e t A be a topologicaI Sidon set of real numbers and dA the infimum of the distances between two distinct points of A. For each I in 10, 2- ldA[ there is a constant C(I), which is an increasing function of I, such that any bounded function b : A [ - I , I ] -+ C which is constant on each interval [A - I, A I] is the restriction to A [-I, I ] of the Fourier transform of a measure whose norm does not exceed C(1) sup IbJ.

+

+

+

A

Remark: The condition 21 < dA ensures that the intervals [A - I, 1 + I], A E A , are disjoint. If 21 = dA, Theorem VI can fail. For instance the set A of all real numbers 2k and 2k 1 k-l ( k 2 2) is a topological Sidon set. For this set dA = 1 and Theorem VI cannot be true for 1 = 3. The function b : A + [-3, $1 + C which is - 1 on all intervals [2k - 3, 2k + $1 and 1 on all other intervals ofA + [-+, $1 is not the restriction to A + [ - 3 , 31 of a uniformly continuous function; hence this function does not belong to B ( A + [-4, $1). The idea of the proof is to discretize the problem, replacing R by n-'Z for a sufficiently large n ; n is an integer large enough for 2/n + 21 < dA and lln < 21, to hold, where I , is defined to be the 1 of Lemma 3. Each interval [A - I,, 1 + I,] contains at least one point A' in n - ' Z ; we arbitrarily choose one such point and let A' be the set of all such A'. Then A' is also a topological Sidon set. For each A in A, let F, be the finite set [A - 1 - Iln, A + I I/n] n n - l Z - A'. Then all the F, are clearly contained in the same finite subset F of n - l Z . The set& + F is a finite union of translates offl'; hence, by Theorem IV, A' + F is a topological Sidon set. The set A" of all A' + FA,A' EA' is contained in A' + F and is a topological Sidon set. It should be noticed that A" is the intersection of n - l Z and the enlarged set A + [ - I - n - l , I + n-'I. The function b can be extended to a locally constant function still denoted by b on the union of the intervals [A - I - lln, A + I + l / n ] A EA.Consider now the restriction b of this new b to A"; b" is the restriction to A" of an element p of B(R), whose norm does not exceed AIlb/I,. Let y : R + C be defined as follows: (a) if t E n-'Z, y(t) = p(t); (b) on each interval [n-lj, n-' ( j + I ) ] , y is linear. ~ , IIlllBcR) the If we can prove that y belongs to B(R) and that I l y l l ~ (= proof is complete: on elements t of n-'Z belonging to the same interval [A - I - l / n , 1 I + Iln], b" is equal to the constant b(A) and so is y.

+ +

+

+

pl,

The norm in B ( R ) of a function pl : R -+ C is the same as the norm of defined by pl,(t) = p, (at). The following lemma now ends the proof:

LEMMA 5: Let g, : R -+ C be any element of B ( R ) and let y : R -+ C be defined by the two conditions: q ( j ) = y ( j )for allj E Z a n d y is linear on each interval j + 11. Then y belongs to B(R) and 11y11~(~) < IIp,llncR). Proof: Let S be the measure ((x - j). By Poisson's formula, the Fourier transform of S is S itself. Let A : R -+ [0, 11 be the function defined by A(t) = sup (0, 1 - Itl). Then A = ti, where u 2 0 belongs to L1(R);A is a positive definite function. Let u(x) = u ( x + k ) be u * S. The Fourier transform of u is the product A S = N x ) . Hence v = 1. With this notation y = (ST) * A is the Fourier transform of the measure Y = ( S * p) u wherep = 9 . We compute the norm of v :

u,

x:",

x::

Lemma 5 is proved and so is Theorem VI. 6. Interpolation of bounded functions defined on a topological Sidon set by the Fourier-transforms of complex bounded Radon measures on R THEOREM VII: Let A be a topological Sidon set and dA the infimum of the distances between two distinct points o f A . Then there exists an increasing function w : [0,2- IdA[ -, (0, + a)with the following property: for each 1 in [O, 2-IdA[, each positive E and each bounded function b : A + C, we can Jind a complex bounded Radon measure p such that (a) on each [A - I, A + 11, A E A, @(t)= b(A)A , ( t - A), where A,(s) = sup (0, 1 - lsI/O; (b) outside A + [-I, I ] , I,A(t)l G s ; This result is striking because, with an error not exceeding s, the extrapolation of b :A + C is completely defined on the whole real line. The proof of Theorem VII is identical to that of Theorem 111: the only difference is the definition of p, and off. Here p, is the measure defined by Theorem V1 such that &(t) = w, on [Aj - I, Aj + El, and f in L1(R) is defined byf(t) = sup (0, 1 - I-lltl).

7. Estimates for trigonometric sums whose frequencies belong to a topological Sidon set THEOREM VIII :Let A be a topological Sidon set of real numbers and K any compact interval [a, b], a < b, of real numbers. Then there exists a constant C > 0 such that for every trigonometricsum P(t) = CAE,c, exp 2ni1t whose frequencies belong to A , we have

xmEn

(xrnea

IP,(t)l < 2"IZ l ~ , ( t ) 1 ~ ) by " ~ Schwarz's We have IP(t)l < inequality. Parseval's relation in L2(Q) then gives I,,,lP,(t)l2 = m2". Proof of Proposition 2 : Assume that there exist an integer m 3 1 , a T 3 0 and an infinite set J of integers such that 0 < I j + , - 3L, < T . Replacing J by an infinite subset, also denoted by J , it may be assumed that each difference 3Lf+, - Aj tends to a limit p,, 1 < k < m. If dA = inf Aj+ - 4 we have ,LA,+, - p, 3 dA > 0. Hence the set B of -oo<j<+m

< k < m , has m elements. The set of all sums Aj + ,LA,, j E J, 1 < k < m , is adjacent to A. Hence there is an integer j, such that, if J' denotes

,LA,, 1

,,

In the proof of Theorem VIII, some additive properties of topological Sidon sets will be needed.

the set of all j in J greater than j,, and A' the set of all sums AJ + p j E J', 1 < k < m , we have lc,l < 2C SUP /PI for all trigonometric

7.1. Additive properties of Sidon sets

sums P(t) = &on. c, exp 2ni1t; this is a consequence of the stability of topological Sidon sets; C is defined by the condition that lc,J < Csup IPl for all P(t) = &, C, exp 2niiZt.

We give here some additive properties of topological Sidon sets; for better results, see [lo] or [49].

2 : L e t A be a topological Sidon set of real numbers, written PROPOSITION as an increasing sequence {Aj; - c~ < j < + CO) of real numbers. Then an integer m > 1 can be found such that lim Aj+,,, - 3Lj = + CO. IjI-r+m

Remark: For m = 1 , this property may fail. The setA of all 2k, k > 0, and all 2k + 1 , k 2 0 is a topological Sidon set and for an infinite set ofj, lj+l - Aj = 1. The proof of Proposition 2 is based on the following lemma: LEMMA 6: Let A be a topological Sidon set. Assume that,for all trigonometric sums P(t) = &,A c, exp 2ni1t, lc,J < C sup I PI. Assume that m

I,,

R

is a positive integer, A a finite set of 2'" elements, B a finite set of m elements and that all the m2" sums a b, a E A, b E B are distinct. Finally assume that A contains A B. Then m < C2. Proof of Lemma 6: Let 9 be { - 1, I)", already used in Section 3. Elements w of 9 are sequences (w, , ..., a,) of + or - 1 . We can put 9 in a ( 1 - 1 ) correspondence with A ; let w -t a(o) be this correspondence. Consider the trigonometric sums P,(t) = w j exp 2nibjt, where B = {bl , . . .,b,) and P(t) = P,(t) exp 2nia(w)t. Then P(t) = - + B c(v) exp 2nivt, where c(v) = 1. Hence m2"

+

+

I,,,

=

Z ~ A +Ic(v)l < C SUP IPI. R B

which gives Lemma 6.

1';

+

We shall prove that sup IPI R -.

<J ~ P ,

R

K

Select 2" points among the 4, j~J', such that the corresponding Aj + Bare pairwise disjoint; let A be the set of these 3Lj and apply Lemma 6. We get m < 4C2 which ends the proof. COROLLARY 1: For each topological Sidon set A, there exists an integer m > 1 such that, for every T > 0, a partition A = Mo U k J A jcan be achieved with the following properties (a) M , is finite; (b) Card Aj < m ; (c) the diferent Aj + [- T , TI, j E J, are pairwise disjoint. Proof: We can clearly restrict our attention to A n [0, +a[.Making the corresponding partition for A n 1- co,01 and enlarging M , , we obtain the corollary. Assume that all 1 in A are positive. The integer m is defined by Proposition 2. There exists an integer k, such that k > k , implies A,+, - 1, > (2m + 1) T . Let J be the set of all j > k , such that Aj - 3Lj- 2 (2 + l / m ) T. For each j in J, let j' be the integer in J which immediately follows j ; then j' < j + m and putting Aj = {Aj, b+l, ...,Ay - we get the required partition of A. 7.2. A lemma due to Zygmund PROPOSITION 3 : Let a be apositive element of L1(R),n an integer 2 2 , and d a positive real number. Let E be the set of all (n, n ) matrices

H=[s(xj-~k)ll<j,k$n

196

SPECIAL SERIES (GROUP-THEORETIC METHODS)

for all increasing sequences ( x ~ j 0. The proof depends on the following lemma: LEMMA 7: Let f be un element of L2(R),and let j s n and d be as in Proposition 3. There is apositive E = E (f,n, d ) such that for every sequence ( c ~ j)G ~n of complex numbers c j f ( x - xj)l12 2 E Proof of Lemma 7 : Assume that a sequence c ~ ,1~<, j g n, k 2 1 can be found such that C",cj,k12 = 1 while for some x ~ ,1~<,j < n, such that x j + ~ . k - X j , k 2 d, we have llgk112 = cj,kf(x - xj,k)llz -+ 0 a s k + + a . We can choose a subsequence such that cj,, -+ dj as k 4 + co ; we have JdjI2= 1 and there is an index I, 1 ,< I < n such that dl # 0. Sincethe L 2 norm is translation invariant, it may be assumed that x,,, + 0 and xjSk-+ tj,where - a < tj < +co. The sum gk = c ~f ( x, ~ xj,,) can be divided into three terms. The first one, called gl involves those xjSkwhich tend to - co as k tends to infinity. The second one called giinvolves those xj,, which have a finite limit and the third one, g z , those xjSkwhich tend to + a .Then

(Ix:

(x:

llx:

z:

z:

lim 1lgk112 = lim 11gk1112 + 11gkZ11f + 1lgk31122,

k++w

k-t+a,

which implies -, 0 as k -+ + co. Hence C:' djf ( x - Ej) = 0, where the dash means that this sum is only taken over finite f j . Taking the Fourier transform, we obtain (x:' dj exp - 2niEjt)f(t) = 0. Hence x:' dj exp -2nitjt vanishes on a set of positive Lebesgue measure and all dj = 0. This contradicts dl # 0 and Lemma 7 is proved. Another equivalent version of Lemma 7 is the following. Let a 2 0 be in L1(R).Assume that a is not the zero function. Then for each integer n 2 2 and each d > 0 there is an E > 0 such that for each trigonometric sum P(t) = cj exp 2niAjt, whose frequenciesil, < A, < . < 2, satisfy - Aj 2 d, ( 1 < j < n - I ) , we have

(lgillz

x:

-

IR

IP(t)Iz a(t) dt 2

2

Icj12.

1

Proof of Proposition 3 : With the preceding notation, let X = (c,, = jRIP(t)12a(t)dt2~llX112. Hence H i s Hermitian and all the eigenvalues of H a r e real and positive; let I , < 1, < ... < In be these eigenvalues and X j , 1 g j < n, the cor= 2 E I I X ~hence ~~ responding eigenvectors. We have &H~x, Ij 2 E and det H = I, In 2 E".

..., cn);thenZHtX = z l g j , k G nZjckii(3ij- A k )

~~jl~~lli i;

7.3. A jirst step in the proof of Theorem VIII First of all, a weak form of Theorem VIII given by Proposition 4 below will be proved:

PROPOSITION4 : Let A be a topological Sidon set and 1 a positive real number. Then there exists afinite subset F ofA such that for all trigonometric sums P whose frequencies belong to A\F, sup IPI and sup IPI are R t-1.11 equivalent norms. We shall prove later that we can choose F = PI. Proof of Proposition 4: Let a(t) be the function sup (0, 1 - It(/l).The idea of the proof is the following: for any b :A + C bounded by 1 , we construct a measure v = ap such that llvll < C and ($(A)- b(A)I < Then v is carried by [-I, I] and Lemma 2 provides the proof. In fact the inequalities - b(2)l < will hold for A €A\Mo, where Mo is a fixed finite subset of A and these inequalities will be obtained by means of a somewhat technical choice of p which we now describe. Roughly speaking, to get p, we write = ,d * 8 ; ,d will be concentrated on a small neighbourhood ofA and a suitable partition ofA enables to separate the different terms of ,d * 8. (1) A partition o f A . Let 8 be the Fourier transform of a, E a positive real number whose exact value will be given at the very end of the proof and T > 0 a real number sufficiently large that jlrl>T I8(t)l dt < E' (in fact > 0). We apply Proposition 2 to get a decomposition of A, A = Mo Uj,z Aj, where CardAj < m and where the Aj $ [-T, TI are pairwise disjoint; Mo is a finite subset ofA. (2) A function d' used in the dejinition of p. Let d = dA be the infimum of the distances between distinct points ofA and let A' : R -+ [O, + a )be defined by A1(x) = k sup ( 0 , l - 31xI/d), where k > 0 is a constant such that j, A'(x) 8(x) dx = 1. (3) A measure p. Let c :A -+ C be a bounded function whose connection with b :A + C will be explained later. The measure L,A has the following three properties : (a) llpll < Cis-l sup A Ic(4I;

+.

+

(b) P(x) = c(2)A' ( X - 2) if I X - A( (c) I,d(x)I g E sup Ic(A)I otherwise.

< d/3, 2 EA;

A

(4) A function c :A + C and a measure dv. If Card A j = 1 and 2 EA j , b(2) = c(A).

198

SPECIAL SERIES (GROUP-THEORETIC METHODS)

If 2 < Card A j < m , we write Aj as an increasing sequence A,, .. .,A, (all of whose terms depend on j), 2 Q k < m, and for 1 < s < k, c (A,) = x, is defined to be the solution of the linear system

LEMMA 8: Let A be a coherent set of frequencies and K a compact set of K If 1 and sup If 1 are equivalent norms for all real numbers such that sup

where b : A+ C is the function that we attempt to interpolate in the Fourier transform of a measure carried by [ - I , I ] and f is the convolution B * A'. Proposition 3 shows that for a constant C2 depending only on I, d and m , we have (x,J < C2,(1 Q s < k)-we recall that Ib(A)l Q 1 for all A EA --, hence IbI Q C ~ / E ; /2(x) = c(A)A' ( X - A) if J X - ill Q d/3 for some A € A ; I/2(x)I Q C2&otherwise. We define dv ( t ) = a(t) dp (t). An estimate for l4(A) - b(A)l. If A € A j and Card Aj = 1, we write $(A) = (d * /2) (A) = jR,d (A - t ) B(t) dt as the sum of three terms I , , I, and I, ; I, = jdi,3,, ,d ( A - t ) B(t) dt, I2 = jIISCI t I /2 ( A - t ) B(t) dt and I3 = j l t l > T / 2 (1 - t ) B(t) dl. Then 1121Q C2c since 1/21 Q E on the domain of integration while jR IB(t)l dt = 1. On the other hand 1131Q C,E since I,dI Q C3/&and j I t I > T IB(t)l dt Q E'. The main term I, = b(A) j!:,3 Ar(t)B(t) dt = b(A). Hence 14(A) - b(A)l < E (C2 C,). If A € A j and 2 Q CardAj < m , we put A = Ajs; Q, is the union of Intervals [A, - d/3, A, + d/3], 1 Q r Q k, Q2 is the complement of Q, lying in [Ajs - T , Ajs TI andQ, is the complement of [Ajs - T , Ajs + TI. We write 4(A) = I , I, I,, where It = So, /2 ( A - t ) &(t)dt For the same reason as above, 1121< C2&and 1131Q C3& On each interval [Ajr - d/3, Ajr d/3], ,A is c(Ajr)A' (X - Aj,) Hence I , = cArf(Ajs - Aj-1 = b(Ajs) In all cases [$(A)- b(A)l Q C4&,where the constant C4 depends only on I and A. Proposition 4 is proved if C4&Q 4.

norms for all almost periodic functions f whose spectra lie in A u (0). Proof: If the lemma were false, we could find a sequence gk = a, of such almost periodic functions with the following properties:

R

almost periodic functions f whose frequencies belong to A. Assume that 0 does not belong toA. For eachpositive E let K, be the set of all real x whose distance from K does not exceed E. Then sup 1f 1 and sup If 1 are equivalent K.

+

+ + +

+

x:=,

7.4. Conclusion of the proof of Theorem VIII To prove Theorem VIII, it remains to add to A\M, the finite set M,. Since Theorem VIII is translation invariant onA, the following lemma can be applied:

(b) sup lgkl R

R

+fk

=

This second condition implies lakl Q 1. Taking a subsequence if necessary, it may be assumed that a, + a as k + + co. Then sup la + fki K.

+ 0 and the sequence of ( f k ) k p is a Cauchy sequence in %(K). Since

sup 1 fkl and sup I fkl are equivalent norms, fk converges, on the whole of R , K

R

to an almost periodic function f such that f = -a on K, and whose frequencies belong to A. For each h. Ihl Q E,f ( x + h) - f ( x ) = 0 on K. Hence f ( x + h) = f ( x ) on the whole of R , since the spectrum of f ( x h) - f ( x )is contained in A, and f is a constant. Since 0 # A there is only one possibility: f = 0. Hence a = 0 and sup Igkl + 0. We thus have a contradiction. R

+

B. Idempotent and semi-idempotent measures 8. Idempotent measures on 1.c.a. groups 8.1. Some definitions: idempotent measures, S(p), the coset ring Let Gbe an1.c.a. group,rthe dual group andM(G) the space of all bounded complex valued Radon measures p on G.

DEFINITION 2: A measure p in M(G) is said to be idempotent if the convolution product p * p equals p. This definition is equivalent to the following one: DEFINITION 3 : A measure p in M(G) is said to be idempotent ifits Fourier transform ,A takes only the values 0 or 1.

Let ,u be an idempotent measure; put S(p) = ( y E T ;P(y) = 1). It is equivalent to determine ,u or S(p). To give the characterization of S(,u) for idempotent measures ,u, another definition will be needed.

r

4: Let SZ be the coset-ring of defined in the following way: DEFINITION SZ is the smallest family of subsets of with the properties: (a) SZ contains all translates of open subgroups of by elements of i.e. open cosets of (b) SZ is stable with respect to finite unions, finite intersections and complements.

r

r;

r

r,

THEOREM IX: Idempotent complex valued Radon measures p on G are characterized by the property: the support S(p) of the Fourier transform ,ii of p belongs to the coset ring of r.

8.2. The trivial half of Theorem ZX PROPOSITION 5 :Each element E of Q is an S(p)for an idempotent measurep. Let A be an open subgroup of and H the annihilator ofA. Then r / A is discrete; hence H , the dual group of r / A , is compact. Let m H be the Haar measure of H normalized by m H ( H )= 1. Then m , can be regarded as an element of M(G) and has the following properties: (a) if y E A then GH(?)= 1 ; (b) if y &Athen &&) = 0. Hence A = S(mH). If E = A + y o , we define dp = xro(x)dmH ( x ) ; then E = S(p). To prove Proposition 5, it suffices to show that the family 9'of S ( p ) for idempotent p is stable with respect to finite unions, finite intersections and complements. If p and v are idempotent measures S ( p * v) = S ( p ) n S(v), S ( p v - p * v) = S(p) v S(v) and S (6 - p) = I'\S(p), where 6 is the unit mass concentrated in 0 and \ denotes complementation. Hence SZ' =, Q. The proof of the reverse inclusion SZ' c SZ is deeper.

r

+

8.3. Reduction to the compact case LEMMA 9: Let p and v be two distinct idempotent measures on G. Then Ilp - vJl 1. The same result holds if the Fourier transforms of p and v take only integral values. - 911, = 1. Proof: Jlp - vIJ 2

DEFINITION 5 : The support group of p is dejined to be the smallest closed subgroup of G on which p is concentrated. PROPOSITION 6: I f p is idempotent, the support group of p is compact. Proof: Let H be the support group of p. Consider p as a measure on H and let I" be the dual group of H. By definitionof H, p isnot concentrated on a proper closed subgroup of H , therefore for all y # 0 in x,,u # p. Hence IIxyp - pIJ 2 1. On the other hand IIxYp - pi] tends to 0 as y tends to zero in I". These two observations imply that T'is discrete. To see that I I x , , ~ - pll + 0 as y + 0, let C, for any positive E, be a . the neighbourcompact subset of G defined by j,,, dip] < ~ 1 4Consider hood V of 0 in I" defined by sup Ix,(x) - 11 < s/211pj1. Then, for any y

r',

K

SK

in V , llxvp - pll < 2 JG\K dlpl + ~I211pIIdlpl < E. Assume now that Theorem I X has been proved for any compact group H. Let p be an idempotent measure on G and H the support group of p ; let A be the annihilator of H a n d set S H ( p )= { y E T/A;@(y) = I), where p is regarded as a measure on H. If n : + r / A is the canonical homomorphism whose kernel is A, S ( p ) = n - I (S,(p)). If SH(p)belongs to the coset ring of r/A,then S(p) belongs to the coset ring of F.

r

8.4. 'Translations lemmas' We arrive now at the main part of the proof. The main tool in the proof (Theorem X below) is a refinement of the translation lemma of Helson ([15],p. 66, lemma 3.5.1). In all of what follows it will be assumed that G is metrizable which implies that the dual group is countable. On M(G) the weak-star topology o (M(G), C(G))is defined by the duality with the space of all complex-valued continuous functions on G. The unit ball Ilpll < 1 of M(G) is a metrizable compact convex set for the weak star topology. The trivial modifications necessary when G is not metrizable are left to the reader: all sequences should be replaced by nets. The following result can be expressed in two equivalent ways; as usual B ( r ) denotes the Banach algebra of Fourier transforms of all complex Radon measures p on G.

r

THEOREM X :Let p be a complex Radon measure on G, (y&*

a sequence

of elements of the dual group T, y,(x) the value at x of the character y, : G + T and dpk(x) the complex Radon measure y,(x) dp (x). Let p = pa p, be the Lebesgue decomposition of p : p a = f dx, for some f E L1(G)andp, is singular with respect to the Haar measure dx of G. Assume that p, tends to a complex Radon measure v in the weak-star topology. Then either lim llv - pkll = 0 ( k -t + oo) or the weak limit v has the following two properties: (i) lvl < Ips] and therefore v is singular with respect to dx; (ii) llvll < Ilpll (note that the inequality is strict). An equivalent way of stating this result will be given. Let v be an element of B(r) and (yk)k2 a sequence of elements of Assume that for each y in r,g, ( y - y,) tends to a limit y(y). Then either lim Ily(y) k++w - y~ ( y - yk)IIBcr)= 0 or y is the Fourier transform of a singular measure and I Y I I B c ~ , < I I v ~ I B V ) . Before proving Theorem X , a corollary must be given.

+

r.

COROLLARY 1 : I f the measure p and the sequence (yk)kal satisfy the i i of p takes only hypothesis of theorem X , and if the Fourier transform , integral values, then either ykp = v for sufjiciently large k or lvl < lpsl and IIvII < Ilpll. Proof: Lemma 9 shows that if ykp = pk is a Cauchy sequence, then p, = v for sufficiently large k.

8.5. Proof of lvl < Ipsl: If ( Y , ) ~ , belongs ~ to a finite subset of T,(p,) belongs to a finite subset of M(G) and for sufficiently large k , p, = v. In this case Theorem X is proved. Otherwise there is a subsequence of the y,, k 2 1, tending to infinity. The following lemma will be useful: LEMMA10: If y k , k > 1 , tends to injinity in r and f E L1(G), then f,(x) yk(x)f ( x ) tends to 0 in the weak star topology o (M(G),C(G)). Proof of Lemma 10: Since 11 fklll = 11 f 1 1 , it suffices to test the weak convergence on a total subset of C(G); for example we can test on all characters y(x), y ET.But in this case jG fk(x) Y(X) dx = f ( y - yk),which tends to 0 by the Riemann-Lebesgue lemma ([15], th. 1.2.4, p. 9). To prove IvJ < lp,I, it suffices to show that, for any g in C(G), g dvl < j~kldlpsl. But g dv = lim ~ G ~ ( x ) Y ~ ( x )=~ Plim( x~)G ~ ( x ) Y & x>

8.6. The inequality (ii) depends on the following two interesting lemmas: LEMMA11 :Let p be a complex valued Radon measure on a compact space G, 8 a Borel function on G such that 161 = 1 and 8p = Ipl, E a positive number in (0, 1) and f a complex valued continuous function on G such that l l f llm G 1. Then We (jGf dp) )P ( 1 - E ) llpll implies llf - 811, G Jg IlpII, where the L1 norm is taken in the space L1 (dJpl)and We denotes the real part. We already know that llpll = sup We jG f dp; Lemma 11 locates Ilfllm~l

precisely those f for which this supremum is approached. Now to the proof. If lzl < 1, 1 - Wez 2 11 - zj2/2.Hence W e j Gf dp plG~~(l =%'eJG f;dlpl)P(l-E) ljpll i m p l i e ~ 2 - ~ ~ ~ I l - ~ f 1 ~ d l-EeOf)dlpl < (Ipll - ( 1 - IIpII < ~llpll.Schwarz's inequalitygives ( J G 11 - Of I dlpl)' < llpll JG 11 - Of l 2 dlpl < 2~llp11'. Since 181 = 1 , the required inequality follows. LEMMA12 : With the hypothesis of Lemma 1 1 , let (gk),>l , be a sequence of Borel functions on G such that lgkl = 1. Assume that gkp tends to a limit v in the weak star topology o (M(G), C(G)). Then either llv - gkpll -t 0 as k -, + a or llvll < llpllClearly llvll < Ilpll. We shall prove that llvll = IIpII implies the strong convergence of g k p to v. For any positive E , let f be a continuous function on G such that 11 f < 1 and W e JG f dv )P ( 1 - E ) IlpII. The weak-star convergence implies that for k 2 k(&),We jG fgk dp )P ( 1 - 2 ~ IlpII. ) Lemma 11 gives ll fgk - 811 < 2 & Ipll ; hence Ilf - g k e l l < 2 & lipll and for k and k' k ( ~ ) ,ll(gk, - gk) 811, = IIgk, - gkII < 4 J E 11.fiII. The sequence gkp is thus a Cauchy sequence and v is its strong limit. Returning to Theorem X , we apply Lemma 12 to gk(x) = yI(x) to terminate the proof.

=

SG

ISG

k++m

x f ( x ) d x + lim jcg(x) yk(x) ~ k++m

lJGg dvl

< JG Ig 1 d ! ~ # l -

k++m

( x )= lim jcg(x) ~ k ( xd~~ ) (x). Hence

P S

k+ C m

8.7. Proof of Theorem ZX From now on, a larger class F(G) of measures p will be used: F(G) is the subset of M(G) defined by P(r)c Z. It will be proved that for each p in F(G) and each rational integer a, the set of all y in such that P(y) = a belongs to the coset ring of r . Sketch of the proof: We have to show that on a compact group G each complex valued Radon measure p with integral Fourier coefficients can be written

r

where mj is the Haar measure of a compact subgroup Gj of G normalized by jGjdmj = 1 and Pj is a trigonometric sum over Gj with integral coefficients. The first term dMl = Pl(x) dm,(x) of this sum will be determined, up to multiplication by a character, by a minimal property: among all weak limits of products y(x) dp (x), y ~ rM1, has the smallest norm. Putting ,u = M, + M i , it will be shown that llpll = llMl[l IIM;II, llMlll 2 1 and 11 M i 11 < llpll - 1. Repeating this descent argument, we get (6.11).

+

r

For a p in F(G), d is the set of all measures y(x) dp (x) for all y in such that jGy(x) dp (x) # 0.All measures in &' have the same norm I(,u\(. Two cases may occur: (1) d is finite; then d = (7, dp, . ..,% dp). Let Go be the support group of p. Consider the Fourier coefficients of p with respect to the dual group of Go: all but n of them vanish. Hence dp (x) = x:pjyj (x) dm, (x), where m, is the Haar measure of Go normalized by mo(G,) = 1 and where the p j are rational integers. Theorem IX is therefore proved in this case. (2) at is infinite. Let 2 be the weak star closure of &' : 2 is compact and metrizable. Since jGdv is a rational integer different from 0 for all v E d , JjG dvl > 1 for all v E 2;hence 2 does not contain 0. The corollary of Theorem X shows that, if p E F(G), each v in 2\&' has the following two properties: llvll < Ilpll and v is singular with respect to the Haar measure of G.

8.9. An optimal measure v The norm decreases as we pass to the weak limits; hence there is a v in d whose norm is minimal. Let y be an element of r a n d ydthe set of all measures y(x) di2 (x) for all measures i2 in d. Since r is a group, y d is contained in 2. Let a be the set of all measures o = yv for which JG do # 0. Then 9? is finite, for otherwise the preceding argument would imply the existence of a o E 3 c d such that 0 < lloll < llvll and this is impossible by the optimal choice of v. We have 9 = -@,v, ..., T,,v), where y,, ..., y, ET.

8.10. DeJinition of the group GI and computation of v If ,L does not assume the value a on I', any translate of ,L also has this property. Since Z is discrete, B which is a pointwise limit of translates of $, does not assume the value a. Let p j = jGTj(x) dv (x), 1 < j < n. Then each p j is a rational integer and is a value assumed by ,L. Let G1 be the support group of v : regarding v as a measure carried by GI, all but a finite number of the Fourier coefficients of v vanish. Hence dv (x) = 2;pjyJ (x) dm, (x), where m, is the Haar measure of Gl normalized by dm1 (x) = 1-

IG*

8.11. Computation of the restriction M I of p to GI in terms of v

r

PROPOSITION 7: There exists a character y' in such that dM, ( x = y '(x) dv (x). The proof of Proposition 7 depends on the following lemma: LEMMA 13: Ifv is the weak limit of the sequence dp, (x) = y,(x) dp (x), then v is also the weak limit (in the topology o (M(G,), C(G,)) of the sequence yk(x) dMl(x). Proof of Lemma 13: We write p = M l -t M i , where M, is carried by GI and where (GI) = 0 TOprove Lemma 13, it suffices to show that ykM', = pCl; tends to 0 in the topology o (M(G), C(G)). For each positive E, there exists a continuous function f : G -t [0, 11 such that JG f l d ~ ; (< E and f = 1 on GI. Then for each g in C(G), IfG fg dp61 < sllgll,. Writing g = fg (1 - f ) g, we get, for sufficiently large k, (jG(1 - f ) gy, dp( < 8, since y, dp tends to v carried by GI on which (1 - f ) g = 0. Since (1 - f ) p, = (1 - f ) p;, we have ISG g dpkl < E (1 + JJgJlm ) for some sufficiently large k. Lemma 13 is now proved. Proof of Proposition 7: We apply Theorem X to the group GI and the sequence of measures y,(x) dM,(x), k 2 1. There are three cases to consider: (a) 11 ykdMl - vl) -t 0 as k + + co, which implies I] MI - 7,vll -+ 0 as k + + co. Since v EF(G), the Cauchy sequence (7,~)is stationary for sufficiently large k and M, = fkv. (b) the set of measuresy, dM, is infinite and does not converge in norm. But then its weak limit is singular with respect to the Haar measure of GI, and this is not the case here. (c) the set of measures ykdMl is finite. Then the weak limit v = ykdMl for sufficiently large k. In all cases dM,(x) = yf(x) dv(x).

+

We have proved that dMl ( x ) = z:pjyS ( x ) dm, ( x ) , where the pj are rational integers and dm, is the Haar measure of the group G I . I t may be observed that d M , is a weak limit of measures y(x) d p ( x ) for y E T , and that (1 M l (1 is minimal.

belongs to the coset-ring of r. Hence there is an idempotent measure o' such that o' p and Theorem XI will be proved. From now on, it will be assumed that p E F+(G).TWOcases may occur.

8.12. At this stage, we write M i = p - M I . Then llMill < 11p11 - 1. Everything that was done with p can be repeated with M i . After a finite number of steps, we get p = M I + + M,, where each M j has the same form as M I . The set of ally for which @ takes a given value a can be defined, starting with the sets E (b,j), where &fj takes the value b, by means of the operations of finite union, finite intersection and complement: write all decompositions a = b, + ... + b,, where bj E Z and lbjl < Ilpll; then the set E(a) of y for which @ takes the value a is the union, taken over all these decompositions, of the intersections E ( b o ,0) nE (b, , 1) n ... n E (b,, m). Now each E (b,j ) belongs to the coset ring of r and so, therefore, does E(a). Hence Theorem IX is proved.

9.1. The periodic case

9. Semi-idempotent measures Let r be a countable subgroup of R; we consider r a s a discrete abelian group. Let G be the compact dual group.

6: A complex radon measure p on G is said to be semiDEFINITION idempotent if for all t > 0 in T, we have @(t)= 0 or @(t)= 1.

-

Let r+ be the set of all y > 0 in r a n d p, :T+ -+ Z the restriction t o r + of the Fourier transform of p. The periodic case is, by definition, the case where a d > 0 can be foundinrsuch that for all y i n P , g, ( y + d ) = &). In this case, a measure o is given by the following lemma: lim [J(x)'Jn dp ( x ) LEMMA 14: With the preceding assumptions, n++m

do ( x ) exists; this limit is taken in the weak topology o (M(G), C(G')) and has the following two properties: o E F(G) and for each y in r + , 8 ( ~=) Lemma 14 clearly gives Theorem XI in the periodic case; d(x) is the complex conjugate of the value taken at x E G by the character d E Hom (G, T). Proof of Lemma 14: Put dp, ( x ) = [a(x)Ydp (x);then llpnll = IIpII and to show that p, has a weak-star limit o, it suffices to prove that @,(y) has a limit for each y in If y > 0, /&(y) = p, ( y nd) = y(y) for all n > 0. If y < 0, there is an integer m such that y + md > 0, as r is a subgroup of R. Hence for n > m , ,G,(y) = p, ( y nd) is constant. =

,w.

r.

+

+

9.2. The aperiodic case THEOREM XI: With the above notation, there exists, for any semi-idempotent measure p, an idempotent measure o such that 8(t) = @(t)for all t > 0 in T. As in the proof of Theorem IX, the hypothesis will be enlarged.

7 : Let p be a complex Radon measure on a compact group G DEFINITION whose dual r is a subgroup of R. W e write p E F+(G) i f @ ( y )E Z for all y > 0 in T. DEFINITION 8 : Let pl and p2 be two complex Radon measures on G. W e = for all positive y in r. write pl -- p2 It will be proved that for each p in F + (G),there is a o in F(G) such that o p. If p is semi-idempotent, the subset E of T,where 8 takes the value 1

-

If the hypothesis of Lemma 14is not satisfied, Ilyf(x)dp ( x ) - yV(x)dp (x)ll > 1 for distinct points y' and y" in r+. We define d to be the set of all measures dv ( x ) = Y(x)dp ( x ) for ally in r+such that JG dv ( x ) # 0; then JG dv ( x ) E Z. If no such y exists, it is plain that p 0 and Theorem XI is proved. Two other cases may occur: (a) d is$nite: Since for distinct y in T f , the corresponding measures dv are different, d is finite if there exist an integer n > 1, n distinct elements y , , ..,y, of r+and n non-zero rational integers p, , .. .,p, such that: if y = y j , 1 < j < n, @(y)= pj and if y # yj for all j, 1 < j < n, &(y) = 0. Then p -- C: pjTj ( x ) dx = o and Theorem XI is proved. (b) d is in jinite: Let d be the weak star closure of d.Then d is metrizable and compact and 0 6 d.

-

.

9.3. An optimal measure v and a number I The measure v is an element of 2 whose norm is minimal. There is a such that dp, ( x ) = Yk(x)dp (x) tends sequence (yk)kg of elements of to v in the weak-star topology. Passing to a subsequence if necessary it may be assumed that yk -t I 2 0 in the usual topology of R (I may be co).

r+

+

LEMMA15: Let g be the set of all measures de ( x ) = Y ( x )dv ( x )for all y > -I such that y E and jG de ( x ) f 0. Then the weak-star closure 3 of is contained in 2. is contained in 2. Writing dv ( x ) Proof: It suffices to show that = lim Yk(x)dp (x) and de, ( x ) = (7 7,) ( x ) dp ( x ) , we get d~

r

+

k++m

dek. Since y > - I and

lim y, = 1for the usual topology &-++a k++w of R, we have y y, > 0 for sufficiently large k and so the measure de, belongs to the set of all measures V f ( x )dp (x) for positive y' i n r . To prove that dpk E a2 it suffices to prove that jG dek f 0 if k is sufficienly large. But jG dek = 0 for an infinite set of k would imply jG de (x) = 0 , which is not true. Lemma 15 is therefore proved. =

lim

From now on, in the periodic case our optimal v will be replaced by the optimal yv, for a y ETsuch that JG yv, # 0. We can then write, in the periodic case, dv ( x ) = P,(x) dm, ( x ) . (2) The aperiodic case; G? is infinite: Since we are in the aperiodic case, Il y'v - y"vll > 1 if - I < y' < y". Let y,v, k 2 1 , be an infinite sequence of distinct measures in G? and let Q be the weak-star limit of an infinite subsequence of these ykv. Thene E d ,Q # 0 and, by Theorem X, Ilell< IlvII which contradicts the optimal choice of v. This case therefore does not occur. (3) The aperiodic case; G? isJinite: This means that, for all y > -I, 9(y) = 0 with the exception of n values of y, y j , for which pj = B(yj) # 0 (1 < j < n). We can then write dv ( x ) = pjyj (x)) d x dv' ( x ) ,where Bf(y) = 0 for ally > -1in.T.

(x:

+

+

9.4. Computation of v LEMMA 16: There exist a closed subgroup G , of G and a trigonometric sum P , with integral coefficients such that dv ( x ) = P,dm, ( x ) dv' ( x ) ,where m , is the Haar measure of G , and the Fourier transform of v' vanishes on I ' n (-1, + a ) . Three cases may occur in the proof: (1) Theperiodic case: There exists a d > 0 i n r such that for each y > - I, B ( y d ) = B(y). We define a measure v , by dv,(x) = lim [ a ( x ) ydv ( x ) ; then v , E F(G), gl(y) = 9(y) for all y > - I in and the product Yv, belongs to d for all y in I' such that jG7 ( x )dvl(x) # 0 , since y + nd > 0 for n sufficiently large. Since the norms decrease in the passage to weak limits llv, 11 < llvll; vl = 0 implies that 9(y) = 0 for all y > -1 (in this case $(y) = 0 for all y > 0). By the optimal choice of v, llvlll = llvll and the norm of each non-zero weak-star limit v, of yv,, y el', also has this property. By the proof of TheoremX, dv, ( x ) = P,(x) dm, ( x ) ,for otherwise dv, ( x ) would be a sum P,(x) dm, ( x ) + P2(x) dm, ( x ) + P,(x) dm, ( x ) and P,(x) dm, ( x ) would be a weak-star limit of products yv, such that 0 < IIPl dm1 ll < llv1 ll - 1.

+

+

r

+

9.5. Computation of p in terms of v From now on let G , be the support group of the measure dv (in the aperiodic case G , = G). LEMMA 17: With the preceding notation, i f P , is not identically 0 , dv is not singular with respect to the Haar measure dm, of G , . Proof: In the periodiccase, we have replaced v by v, = P , dm,, which is absolutely continuous with respect to the Haar measure of G , . In the aperiodic case we have B(y) = 0 for all y > - I in r with the exception of y = y j , 1 < j < n, for which B(yj) = p j . Then the following proposition will be applied ([15],th. 8.2.3, p. 200). PROPOSITION 8 : Let be a subgroup of R, G the compact dual group and o a singular complex measure on G such that 6(y) = 0 for a l l y > 0. Then jG do = 0. Proof: Let E be the closure, in the Hilbert space L 2 (do), of the vector space of all trigonometric sums P(x) = c o ayy(x). Assume for a moment that 1 belongs to E. This means that we can find a sequence P,(x) = a,(y) y(x), such that lirn j, 11 PnI2dlol = 0. Hence, by

r

zr

n-r

Schwarz's inequality,

1,

lirn n++m

+w

jG (1 + P,)

+

do = 0. But

jG P, do = 0

by

hypothesis; hence do = 0. To prove that 1 belongs to E, let p, be the orthogonal projection of 1 on E. Then 1 - p, is orthogonal to all elements y of E; applying this remark W ( X )= Y ( X )(l - ~ ( ~for 1 )y < 0 , we get jG y(x) I 1 - v(x)l2 (x) = 0. By complex conjugation this is also true for y > 0. Hence ( 1 - p,12

d(ol = cdx; since o is singular, c = 0 and g, = 1. Proposition 8 is thus proved. In our context, we can order the real numbers yj, (1 < j< n) in such a way that y, > yj for all other j. Then, if do (x) = y,(x) dv (x) is singular, B(y) = 0 for all y > 0 and jG do = p, # 0. This is impossible and so v is not singular with respect to dx. Lemma 17 is therefore proved.

9.6. Final stage of the computation of p Let M I be the restriction of p to G, and let p = M, + M i . If v is the weak-star limit of the sequence Vkp,then v, carried by G, ,is also the weakstar limit of the sequence V,M,, k > 1 (Lemma 13). It will be proved that VkM, = v for sufficiently large k ; hence by o, = lim y,v, (v, = P, dm,); we have Lemma 16, M, = lim y,v k+ + m k+ + m replaced y, by a subsequence if necessary, to ensure that the latter limit exists. Note that o, E F(G). Then M, E Ff(G), M i E Ff(G) and since llM',ll < llpll - 1, we reach the required conclusion ,u a, + ... + a, = o E F(G) after a finite number of steps. To show that v = ?MI, y > - 1, Theorem X will be applied. Two cases may occur. +a,we have IIM, - y,vll -+ 0 ; in the (1) If IljjkM1 - vll -+ 0 a s k periodic case v = v, = P,(x) dm, and y,v, k 2 1, cannot be a Cauchy sequence without being constant. Hence M, = ykvfor sufficiently large k. In the aperiodic case Ilyl'v - y'vll > 1 for y' # y", and the conclusion is the same. (2) If v = lim ykM, in the weak-star topology but not in the strong

-

Proof: The notation of the proof of Theorem XI will be used: a2 is the set of all measures X,(x) dp (x); if a2 is finite, A,,, - A, assumes a finite number of values and since A, - k tends to 0, A,+, - A, = 1 for sufficiently large k, which implies A, = k for sufficiently large k. If &is infinite, the optimal measure v is a weak limit of a subsequence of X,(x) dp (x). If x = A,, - A, for some fixedj and sufficiently large k in the subsequence, $(x) = 1. Otherwise $ = 0. Hence the support S(v) of 9 is contained in Z and v is idempotent. Let G, be the support-group ov v and let M, be the restriction of p to G, . Then dM, (x) = A,(x) dv (x) for infinitely many k. Hence S(M,), the support of M,,is contained in Z. Let M', = p - M I . Then S(M;) cAuZ. Repeating the preceding argument, we obtain a decomposition ,u M I + ... + M,,, of p, where the spectrum of each M j is contained in Z for

-

1 < j < m - 1 and the spectrum of M, is finite. Hence A, = k for sufficiently large k.

N

-)

,++a,

topology, v is singular with respect to the Haar measure of GI. We have seen that this is not the case. 9.7. A particular case be an increasing sequence of positive real num9 : Let PROPOSITION bers such that A, - k + 0 as k + co.L e t r be the subgroup of Rgenerated by the A,, k 2 1, and let G be the compact dual group. Assume that a complex Radon measure p on G has the followingproperties: $(A,) = 1for all k 2 1 and,ii(t) = 0 ift > 0 does not belong to the set d OfAk, k 2 1. Then A, = k for sufficiently large k.

+

10. Behaviour at infinity of special a-periodic trigonometric series THEOREM XI1 : Let (A,),, be an increasing sequence of positive real numLetA be the set of all A,, k 2 1 bers such that A, - k + 0 as k -+ +a. and V, the space of all mean periodic functions whose spectra are contained in A. If A, # k for an infinite number of k > 1, there is a mean-periodic function f whose spectrum lies in A which is not bounded. Proof: If all mean periodic functions in VAare bounded, A is a coherent set of frequencies. Hence a positive 1 can be found such that the Banach algebras B (A [- 1,1]) and B (A x [-I, I]) are isomorphic (see Chapter IV, Theorem X). For each E > 0, let A, be defined by A,(t) = sup (0, 1 - Itll~).Then A, belongs to A(R) and llA,llA(R) = 1. Applying TheoremX of Chapter IV, we find a constant C and, for each E in (0, I), a complex Radon measure p, on R, such that Ilp,ll < C and (A, + t) = A,(t) if k 2 1 andltl G I . Replacing 1 by a smaller number if necessary, it may be assumed that 1 < 3.Now assuming this, let T,(t) be the I-periodic function defined on [-5, +] by Tl(t) = 2AZl,,(t) - A1,,(t); then T,(t) = 1 on [-113, 1/31 and T(t) = 0 outside [-2113, 21/31. The Fourier coefficients of Ti are 0(k-2) and so TI is the Fourier-transform of a discrete measure o carried by Z.

+

P,

For k 2 k,, lAk - k( < 116. Choose E in (0,1/6); then v, = p, * o has the following properties (i) llv,ll < C2, where C2 is independent of E ; (ii) if t 2 Ako and It - 3Lk( d E, then P,(t) = A,(t); (iii) for all other t 2 Ako7 &(t) = 0. Let I' be the discrete group generated by A. Let j :I'-t R be the canonical injection and h: R + G the dual map. Then h is injective and h(R) is dense in G. When R is thought of as a dense subgroup of G, the bounded measures v, become complex Radon measures on G. In the weak-star topology (a(M(G), C(G)), a subsequence of the v, tends to a measure v with the following properties: (i) llvll < Cz, (ii) 5(t) = 0 for t 2 ilk,, t # A,, and (iii) $(Ak) = 1 if k 2 k,. Proposition 9 shows that ilk = k for sufficiently large k, and Theorem XI1 is thus proved. Letf be an unbounded mean periodic function whose spectrum lies in A. We write f(t) a kexp 2niAktwhere it may be assumed that all the Ak are different from zero. Then (ak12< + co and the primitive g off has the absolutely convergent series (ak/2niilk)exp 2ni3Lkt;hence g is almost periodic and continuously differentiable; the spectrum of g lies in A but the derivative of g is not bounded.

- xka

x,, xk2

11. Notes Helson sets are the compact versions of Sidon sets and are defined by A(E) = C(E). The family of Helson sets is stable with respect to finite unions [94]. Theorem IV is a remarkable discovery of Drury [40] and Theorem VIII was found by Mme Deschamps-Gondim [38]. A simple proof of P. J. Cohen's theorem (Theorem IX) was obtained by It8 and Amemiya [53] and the characterization of semi-idempotent measures can be found in [52].

CHAPTER VII

PISOT NUMBERS A N D SPECTRAL SYNTHESIS

Let 0 > 2 be a Pisot number and E the compact set of real numbers of Cantor type constructed with the dissection ration 0-l. Then for each bounded continuous function pi :R -t C whose spectrum lies in E, there exists a sequence y,, k 2 1, of finite trigonometric sums whose frequencies belong to E such that pi, -t uniformly on each compact subset of the line and sup [pik[ d sup IyI. This is a strong form of spectral synthesis. R

R

In Sections 1-4 classical facts about spectral synthesis are recalled. In Sections 6 and 7 a fundamental theorem is proved, and in Section 8 a p-adic version of the preceding results is given.

1. Spectral synthesis and structure of closed ideals of a group algebra 1.1. The Banach algebra A(R)

Let A(R) be the Banach algebra of all continuous functions f : R -t C, which are the Fourier-transforms of functions in L1(R); A(R) is normed by Ilf I~A(R) = jzz I ~ ( ~ dl.) I Clearly 11 f (1, = sup If ( < 11 f [IA,) and (I f / I A cR) is translation-invariant. R

PROPOSITION 1: Each compactly supported continuously dzfferentiable function f : R -t C belongs to A(R). Proof: The hypothesis implies that f and f ' belong to LZ(R). Let F be the Fourier-transform o f f ; then the Fourier-transform o f f ' is - 2nixF and, by Parseval's relation, F and x F belong to L2. Schwarz7inequality dx)"' < + m and jlxlsl[F(x)~dx gives IF(x)l dx d Ji (j!, IF(" d (llxl, x-' dx)lIZ(jIxI, x21F12dx)lI2 d + m. Hence F belongs to

The following result shows how approximate units can be found in A(R) (A(R) does not have a unit).

214

PISOT NUMBERS A N D SPECTRAL SYNTHESIS

PISOT NUMBERS A N D SPECTRAL SYNTHESIS

215

THEOREM I: Let f and g be two functions in A(R). I f f ( 0 ) = 1, lim Ilg(x) - f(n-'x) g ( x ) l l ~ (=~ )0.

is the restriction to E of a function g of A(R), and the norm off in A(E) is the infimum, taken over all such extensions g, of llgllA(R).

If

PROPOSITION 2: I f f is a.function in A(R) vanishing at 0, the norm o f f in the restriction algebra A [-E, E] tends to 0 with E. Proof: Let or : R + [O, 11 be a function in A(R), which is 1 on [- 1. 11. Then, with E = n-l and the notation of Theorem I, Il f llAc-e, r l < 11 f ( x ) or (nx)llAcR)+ 0 as n -+ co. Since for each f in A(R), Il f llA[-&,&] decreases with E , the general case follows.

n++m

= 0,

lim

n++m

Il f (nx)A x ) I I A ~ R )

=

0.

Let F and G be two functions in L1(R)whose Fourier transforms are f and g and for each n 2 1, let Fn = nF (nx) and pn = n - lF (n- ' x ) . Then llFnlll = IIFlI . We have to prove that

,

In each case if we replace F by F' and G by G' such that ]IF - Ff 11 < E, IIG - G'II < E and if we put Fi = nFf (nx), we get /IFn- Fill < E and, for all n # 0, I(G - F, * G - G' FA * G'II < E ~ l l F l l ~ EIIGII + E~ while lipn* G * GfII1< ~ l l F / l ~EIIG,11 E ~ Hence . it suffices to prove the result for F and G belonging to a dense subspace of L1(R):we choose the continuous functions with compact support. If the supports of F and G are contained in [-T, TI, the support of G - F, * G is contained in [-2T, 2T] and, putting en = sup Ix-ylGn-1T IGb) - G ( Y ) ~ ,we get IG(x) - (F, * G) (41 = ~ j z ~G(x) : ~F,(y) ~ dy - jT1gInG(x- y) F,(y) dyl < ~,,llF,1 1 . Hence IIG - Fn * GII1 + 0 for these continuous compactly supported F and G and the proof of (7.2) is complete. Making a simple change of variables, we see that * GII, = /IF * G,IIl and returning to the case of continuous compactly supported F and G, we get, with the preceding notation,

+

+

+

+

+

+

3 :Let f be a function in A(R) and x a real number such that PROPOSITION f ( x ) # 0. There exists a function h in A(R) such that f ( x ) h(x) = 1 on a neighbourhood of x. Proof: By translation we reduce the problem to the case x = 0 and we may assume that f(0) = 1. If E < 1 , using the notation of Proposition 2, we get 111 - f llA[-c,d = 1101 - f II~c-e,el-* 0 with 8. For sufficiently small E, let g be a function in A(R) such that llgllAm < 4 and 1 - f = g on [ - E , E ] . Put h = x z g k ; this series isconvergent in A(R) and also for the uniform ncrms. Hence hf = 1 on [ - E , E]. PROPOSITION 4: Let K be a compact subset of R and f a function in A(K) which never vanishes on K. Then l/j'belongs to A(K). Proof: Let Y be the algebra of all complex valued functions all of whose derivatives, including the function itself, decay rapidly at infinity; Y is invariant under Fourier transform and is contained in A(R). For each x in K there exists a neighbourhood V(x)of x and a function g, in A(R) such that fg, = 1 on V(x).Standard arguments on partitions of unity show that there exist an integer n, n points x,, ..., x, of K and n regular functions or,, ...,a, in Y such that, writing V j for V ( x )and gj for g, if x = x j , each + is supported by V j and a , + ... + a, = 1 on K. We have: 1 = aj = ajgjf on K and llf = ajgj E A(R).

z: z:

x:

1.2. The ideals I(E) and J(E)

If Eis a closed set of real numbers, the restriction algebra A(E) is defined b y the following rule: a continuous functionf : E -,C belongs to A(E) iff

Among the closed ideals of A(R) ,a first type can be defined in the following way: let E be a closed set of real numbers and I(E) be the set of all f in A(R) which vanish on E ; I ( E ) is plainly a closed ideal of A(R). Let J,(E) be the set of allf in A(R) which vanish on a neighbourhood of E where the neighbourhood depends on f. Clearly J,(E) is an ideal of A(R) and so is the closure J(E) of J,(E) in A(R).

216

PISOT NUMBERS AND SPECTRAL SYNTHESIS

D E F ~ N ~1~: The O Nzero set of an ideal I of A ( R ) is the largest closed set on which all the,functions of I uanish. THEOREM I1 :Let I be a closed ideal of A(R) and E the zero set of I. Then J(E) c I c I(E). The second inclusion being obvious, it suffices to prove the first one. The proof depends on two lemmas: 1 : Let I be an ideal of A(R), E the zero set of I and K a compact LEMMA set of real numbers disjoint from E. Then there exists a function f in I which is identically 1 on K. Proof: For each x in K, there is a function f, in I such that f,(x) = 1, since otherwise x would belong to the zero set of I. Since I is an ideal, g, = f,f, belongs to I ; g, B 0 and there is a neighbourhood V ( x ) of x such that g, > -$ on V(x).Let V , , ..., Vnbe a covering of Kwith n of these V ( x )and let f be the corresponding sum gj where we write gj for g, and V jfor V ( x )if x = x j . Then f E I, and f ( x ) 2 for all x in K. Proposition 4 shows that there exists a function h in A(R) such that f h = 1 on K. Hence fh is the required element of I.

x:

PISOT NUMBERS AND SPECTRAL SYNTHESIS

217

2. Spectral synthesis and atomization of distributions 2.1. We begin with a technical lemma: 3: For any closed set E of real numbers, Y n Jo(E) is dense in LEMMA J(E>. By Lemma 2 it suffices to show that each function f i n A(R) whose support is a compact set K disjoint from E may be approached by a sequence f k of functions in Y. Since Y is dense in A(R), we can find a sequence g, in Y whose limit is$ Let ol be any function in 9which equals 1 on K and which is 0 on a neighbourhood of E. Then the sequence f k = ag,, k > 1 , tends to f as required. 3 : Let P M ( R ) be the space of all distributions S on R whose DEFINITION Fourier transforms 3 belong to Lm(R). We put llSllpM= 1115'11m and this norm is called the pseudo-measure norm. The duality between PM (R)and A(R) is dejined by

+m ( S

P)

=

(9, F )

S ( t ) ~ ( tdt,) F E L 1 ( ~ ) ,S E L'(R).

=

-m

2 : Let I be an ideal of A(R). The compactly supported functions LEMMA of I are dense in I. Proof: If ol is any compactly supported function A(R) such that a(0) = 1, then, for each f in I,fn = f ( x ) a ( n - l x ) belongs to I and tends to .f. Proof of Theorem I1 : Since I i s closed, it suffices to show that Jo(E) c I. Lemma 2 shows that we can further restrict our attention to compactly supported functions g in J,(E). Let K be the support of such a g ; K does not intersect E and by Lemma 1 there exists a function fin Iwhich equals 1 on K. Then g = g f belongs to I and the proof is complete. 2: A closed set E of real numbers is called a set of ~ynthesis DEFINITION if one of the following two equivalent statements is true: (a) each function f in A(R) which vanishes on E is the limit, in the norm of A(R) of a sequence f , of functions in A ( R ) which vanish on neighbourhoods Q k of E; (b) I(E) is the unique closed ideal of A(R) whose zero set is E. Theorem I1 shows that (a) e (b).

For any closed set E of real numbers, let PM ( E ) be the space of all S in PM ( R ) which are supported by E. LEMMA 4: For each S in PM ( E ) and each f in J(E), ( S ,f ) = 0. By continuity we may restrict our attention to f E Y n J,(E), and in this case ( S , f = 0 by the definition of the support of a distribution.

>

2.2. The weak-star topology o (PM(R),A(R)) In what follows o (PM ( R ) , A(R)) will denote the weak-star topology defined by the duality between A ( R ) and PM (R). A base of neighbourhoods of 0 for o (PM ( R ) , A(R)) is given by the sets V (P, E ) defined as follows: for each E > 0 and each finite subset F of A(R), V (F, E ) is the set of all S in PM (R) such that sup I(S, f )I < E. F

Since A(R) contains a contable dense subset, the restriction of the weakstar topology to the unit ball B of PM ( R ) is metrizable and B is compact for o (PM ( R ) , A(R)).

218

PISOT NUMBERS AND SPECTRAL SYNTHESIS

2.3. A description of P M (E) THEOREM I11: Let E be a closed set of real numbers. The following three properties of E are equivalent: (a) E is a set of synthesis; (b) for each S in P M (E) and each f in A(R) vanishing on E, ( S ,f ) = 0; (c) each distribution S carried by E whose Fourier transform is bounded is the weak star limit of a net (pa),,, of measures carried by finite subsets of E. Proof: To get the equivalence between (a) and (b), we consider the ideal J(E) as a closed linear subspace of Z(E). The Hahn-Banach theorem shows that Z(E) = J(E) if and only if (b) holds. The equivalence between (b) and (c) is also given by the Hahn-Banach theorem applied to P M (E) topologized by the weak-star topology o (PM (E), A(R)); let Mf(E) be the linear subspace of all measures carried by finite subsets of E. Then Mf(E) is dense in PM(E) if and only if (b) is true.

2.4. The dual of a restriction algebra DEFINITION 4: For a closed set E of real numbers, we define N(E) to be the subspace of all S in P M (E) such that (S, f ) = 0 for all f in Z(E).

5 : The dual space of the restriction algebra A(E) is N(E). PROPOSITION The proof is trivial.

PISOT NUMBERS AND SPECTRAL SYNTHESIS

(b) yk + g, uniformly on each compact subset of R, then y is bounded and continuous and the spectrum of y also lies in E. In fact let S be the Fourier-transform of g, and let F be any function in L1(R) whose Fouriertransform f vanishes on E. By Lebesgue's dominated convergence theorem

but plk(x) is a finite sum yk(-

X)

F(x) dx

xnGE a (k, A) exp 2niAx, and

=

A2

3. A strong form of spectral synthesis In certain cases that we shall now describe, the property of spectral synthesis gives a very concrete description of the space of bounded continuous functions pl : R -t C, whose spectra lie in a compact set of real numbers. 3.1. We begin with a very simple remark. If E is a closed set of real numbers and if a sequence yk, k 2 1, of trigonometric sums whose frequencies belong to E satisfies the two conditions

a (k,A) P(A) = 0

since f vanishes on E.

Hence (S, f ) = 0 and the spectrum of v, which by definition is the support of its Fourier-transform S, is contained in E.

3.2. Is it possible that, conversely, any bounded continuous g, : R -t C whose spectrum lies in E can be built up as the limit, in the sense of uniform convergence on compact sets, of a uniformly bounded sequence of finite trigonometric sums whose frequencies lie in E? THEOREM IV: Let E be a closed set of real numbers. The following three statements are equivalent: (a) for a constant C > 0 and each bounded continuousfunction g, : R -, C whose spectrum lies in E there exists a sequence v k ,k > 1, offinite trigonometric sums whose frequencies belong to E such that SUP1 ~ k G l c 1 ~ 1 R

1 : A closed set E of real numbers is a set of synthesis if and COROLLARY only if the dual of A(E) is PM(E).

219

R

(7.4) and vk(x) -t q(x) uniformly on each compact subset of Rr (b) for a constant C > 0 and each S in P M (E), there exists a sequence S k , k > 0, o f measures supported by finite subsets of E and such that, for each f in A(R)

(c) each S in P M (E) is the weak star limit of a sequence Skof measures carried by finite subsets of E. Moreover, the same constant C may be chosen in (a) and (b). 3.3. This theorem is surprising since (a) is very precise (for uncountable compact sets we cannot get better approximations) and (c) seems to be fairly general. However, the proof is very simple and uses only standard functional analysis.

220

PISOT NUMBERS AND SPECTRAL SYNTHESIS

(c) => (b) : Let B be the unit ball (ISIIPM < 1 of P M (E) and, for each

Q

> 0, let QBbe the homothetical ball llSllpM< Q.

The Banach-Steinhaus theorem shows that, if (S,, f ) -+ <S,f ) for < + a. each f in A(R), then sup llSkllPM

PISOT NUMBERS AND SPECTRAL SYNTHESIS

Ch. XVI, th. 4.19) the square mean

22 1

~ l,d12 dx is 0 and $ lim 1 1 2 STT T+

+m

cannot be almost periodic. Hence, if E is not countable, (a) cannot be improved to uniform convergence on the whole of R.

k> 1

Since A(R) contains a countable dense subset, the topology o (PM (R), A(R)) restricted to B is metrizable, and B is compact in this weak-star topology. Let B1 c B, be the set of all complex measures ,u carried by finite subsets of E whose pseudo-measure norms do not exceed 1;let C be the weakstar closure of B1: C i s clearly contained in B. By (c) and our first remark, B is the union of the increasing sequence D,, = B n nC, n ,) 1, of compact subsets of B. Since D, - D, c D,, ar,d B is compact, Baire's theorem shows that D, = B for n sufficiently large and (b) is proved. (b) (a) : For each real t , let p,, :E -+ C be defined by yt(x) = exp 2zitx. The mapping @ : R -+ A(E) defined by @(t) = p,, is continuous as (7.1) shows and, for each T > 0, the subset K of all p,, in A(E), for which It1 < T , is compact We apply the well-known result PROPOSITION 6: Let X be a Banach space and L,, k ,) 1, a sequence of continuous linearforms on X. Thefollowing three statements are equivalent: (i) for each x in X,L,(x) has a limit ay k -, + co; (ii) there is a constant C > 0 such that ((L,I ( d C and Lk(x)has a limit for all x in a total subset of X; (iii) L,(x) converges uniformly on compact subsets of X. (a) => (c): By Lebesgue's dominated convergence theorem, (7.4) implies yk(x)f(x) dx -+ jR y(x) f(x) dx for each f in L1(R), which is (c).

SR

3.4. We say that E is a set of strong synthesis if E possesses one of the three equivalent properties of Theorem IV. Varopoulos [94] gave an example of a set of synthesis which is not a set of strong synthesis, and Malliavin [61] gave an example of a closed set E of real numbers which is not a set of synthesis; we shall give another example in Chapter VIII. A countable compact set E is always a set of strong synthesis. Any bounded continuous p :R -+ C whose spectrum lies in E is, in this case, an almost periodic function and the approximation of p, by finite trigonometric sums whose frequencies lie in E holds in the norm of Lm(R). If E is not countable, E carries a diffuse bounded Radon measure ,LA, i.e. p has no discrete part; we put p,(x) = ,d(-x). By Wiener's theorem ([24],

4. Spectral synthesis and weighted approximation Using weight functions, a strong form of spectral synthesis can be defined where the approximation of the function by trigonometric sums holds in the strong sense.

4.1. L1(R, w), L" (R; w), C (R; w), Co (R; w) In what follows w : R -+ [l, + co[ is an even function which is increasing and concave on [0, + a [ , and we assume that w(0) = 1. For example o(x) = 1 + Ixla, 0 < a < 1, is such a function. We define L1 (R; w) to be the subspace of all f in L1(R) satisfying I f(x)l w(x) dx < + a ; this integral is equal to the norm o f f in L1 (R; w). If multiplication is defined to be convolution, L1(R; w) becomes a Banach algebra and A (R; o ) denotes the function algebra of all Fourier transforms of elements of L1(R; w). The dual space of L1 (R; w) is the Banach space of all complex valued Lebesgue-measurable functions a, on the line such that

l?:

Iy(x)l

< Cw ( x )

almost everywhere.

(7.5)

The norm of y in L a (R; w) is the infimum of all the C > 0 which can appear in the right hand side of (7.5). C (R; w) is the subspace of all continuous functions g, in Lm (R; w) and Co (R, w) is the subspace of all p in C (R; o ) such that p,(x)

=

o(w(x)) as 1x1 -+ +co.

(7.6)

4.2. Strong synthesis for weighted norms DEF~NITION 5: The spectrum of a function g, in Co ( R ;w) is the support of the Fourier transform S of p,. Note that g, is a tempered distribution. DEFINITION 6: A closed set E of real numbers has the property S(w) i f each continuous function p, : R -, C which satisjes (7.6) and whose spectrum lies in E is the limit, in the norm topology of Co (R ;w), of a sequence of trigonometric sums whose frequencies belong to E.

222

PISOT NUMBERS AND SPECTRAL SYNTHESIS

PISOT NUMBERS AND SPECTRAL SYNTHESIS

PROPOSITION 7 : If E has the property S(w)for all weights w, E is a set of synthesis. Proof: Let f be an element of L1(R); it is quite easy to construct a weight function w such that lim w(x) = + co and f E L1 (R; o). Let

bounded and if g, is a complex valued continuous function whose spectrum lies in E and which vanishes at infinity, (b) implies that sup R Ig,(x)

- y,(x)I 4 0 as k -+ + co; hence g, is almost periodic and this implies g, = 0. Since we have already proved this property of uniqueness in Chapter 111, we may assume that w(x) + + co with x Theorem V does not give a method of obtaining cpk in terms of g,. In the following statement cp, depends linearly on g, but we lose a little at (c).

Ixl-r+a,

p, be any bounded continuous function whose spectrum lies in E ; then p, E C,, (R; w), and S(w) gives a sequence

p,,(x)

a ( k , A) exp 2niAx

= AsE

such that

(7.7)

THEOREM VII: Let 8 be a Pisot number. There exist apositive constant C and a sequence L,, k 2 1, of linear mappings such that for any bounded continuous g, : R -+ C whose spectrum lies in E, (a) the frequencies of Lk(g,)belong to afinite subset F, of E; (b) I(Lk~) - dx)l Ce-klxl sup R 1~1;

Sincef E L1(R; w),(7.8)impliesjf: v,(-x) f(x) dx+ JT,"p,(-x) f(x)dx. Iffvanishes on E w e get j?: cp(-x) f(x) dx = 0 and E is a set of synthesis by (b) in Theorem 111.

(c) sup lLk~l R

5.1. The results Let 8 > 2 be a Pisot number a n d E the compact set of a l l s u m ~ ~ ~ ~ , O - ~ , = 0 or 1; E is a Cantor set constructed with the dissection ratio 8-'. The following theorems show that very nice forms of spectral synthesis hold for E.

E,

THEOREM V: Let 8 be a Pisot number and let v : R 4 C be any bounded continuous function whose spectrum lies in E. There exists a sequence vk, k 2 1, offinite trigonometric sums such that (a) for each k 2 1, the frequencies of p,, belong to E; (b) p,,(x) -+ y(x) uniformly on each compact subset of R; (c) l ~ k -* l (PI.

5.2. Sketch of the proofs Theorem VII will be an easy consequence of the atomizing properties of harmonious sets found in Chapter IV. By simple modifications jn the proof we shall get Theorem VI. To obtain Theorem V, we shall give in Section 7 a characterization of the restriction algebra A(E) in the style of Bochner's theorem ([15], p. 32, th. 1.9.1).

5.3. Proof of Theorem VII

R

EkaO

The proof will in fact show further, that the frequencies of cp, belong to the finite set of the 2, sums F ~ O -EJ~ ,= 0 or 1.

1:

THEOREM VI: Let 8 be a Pisot number, let co : R + [I, + a [ be an even unction which is increasing and concave on [0, + a[and let p, : R -+ C be a continuous function satisfying y(x) = o(w(x)) as 1x1 + + co. There exists a sequence p,, k 2 1, offinite trigonometric sums such that,for each k 2 1, (a) the frequencies of'p,, belong to E; Theorem VI simultaneously gives synthesis and uniqueness: if w is

c sup 1411. R

In fact, a little more will be proved: there is a finite subset F of E such that F, is thefinite set of allsums C: ~ ~ 0+- 8-,A7 j cj = 0 or 1, A E F; we note further that F, is contained in E.

5. Pisot numbers and spectral synthesis

R

223

1

Let A be the set of all finite sums Ekek,E , = 0 Or 1 and, for each k 2 1, let A, be the subset of the 2, sums 1:.@j, ej = 0 or 1. For each rational integer p, let zp be the isometry of LW(R)which carries p, into the function defined by x -+ p, (Opx). If the spectrum of 9 lies in E and if p = k 2 1, the spectrum of z,(v) lies in OkE= A, + E c A + E. The dilation x -+ Bkx is suggested by the right-hand side of (b). We recall Theorem XI of Chapter IV: let A be an harmonious set of real numbers and E a compact set of real numbers. There exist a finite subset F of E, a linear mapping L which maps bounded continuous functions g, : R -+ C whose spectra lie in A + Einto almost periodic functions

224

y

PISOT NUMBERS A N D SPECTRAL SYNTHESIS

PISOT NUMBERS A N D SPECTRAL SYNTHESIS

spectra lie in A + F and a positive constant C such that: for each il in A, the spectrum of y lies in il + F whenever the spectrum of p, lies in il + E.

= L(y) whose

For a function win the class Y of Schwartz, (7.11)can be improved to sup I Y IQ SUP I W Y I and (7.10) to I ~ ( x-)V ( X ) I < 1x1SUP I W Y I . (7.12) R

R

R

Starting with a y whose spectrum lies in E, the spectrum of t k y lies in A , + E c A + E. If 8 is a Pisot number, A is harmonious. We define Lk(y) to be ( z - , 0 L 0 z,) (91); (a) shows that the spectrum of ( L 0 7,) ( y ) is contained in A, F; hence the spectrum of L,(p,) is contained in O-,A, O-,F = F k . Statements (7.10) and (7.11) give the corresponding statements in Theorem VII.

+

+

5.4. Proof of Theorem V1 A first step in the direction of Theorem VI is the following Proposition 8. PROPOSITION 8: For each Pisot number 0 > 2 there exist a constant C > 0 and a sequence L,, k 2 1 , of linear mappings with the following properties: (a) for each continuous function p, in L m ( R ;o) whose spectrum lies in E, L,(y) is a trigonometric sum whose frequencies belong to E; (b) L,(f) +f uniformly on each compact set of real numbers; (c) I(Lk(f )>lo/ < Iflo/. R

R

We first show why Proposition 8 implies Theorem VI. If y ( x ) = o(o(x)), as 1x1 -+ + co,there exists a weight function s in the sense of Section 4.1 such that p,(x) = 0 ( a ( x ) )and D(x) = o(o(x)),as 1x1 -+ + a.By Proposition 8, I(Lkf)(x)l < CQ ( x ) for each real x ; this inequality and (b) together imply Theorem VI. 5.5. Proof of Proposition 8 Some properties of the weight function o will be needed. We put o ( x ) = 1 a(x); on [0, +a[, a is increasing, concave and a(0) = 0. It follows that a(x)/xis decreasing on [0, + co[ and

+

0 Q s Q 1 and t 2 0 imply a (st) 2 sor ( t ) s a l and t 2 0 imply a (st) Q sol ( t ) s 2 0 and t 2 0 imply a ( s t ) Q a(s) + a(t).

+

(7.13) (7.14) (7.15)

225

LEMMA 5 :Let E be a compact set of real numbers. There is a constant Cl such that,for each continuousfunction y : R -+ C in L" ( R ;o)whose spectrum lies in E

Proof of the lemma: By (7.14) we have 1 < o ( x ) < 1 + C l x ( ;hence if F E L1(R) and xF E L1(R),F belongs to L1 ( R ;o). Taking the Fourier transforms we see that iff E A(R) and f ', the derivative off, also belongs ) Ilf I ~ A ~ R ) + C l l f l l l ~ c ~Let ). to A@), then f A ( R ;o) and I l f l l A ( ~ ; r n6 a : R + [O, 11 be an indefinitely differentiable function which is 1 on a neighbourhood of E and 0 outside a compact subset of R. Put fx(t) = (exp 2nitx - 1) a(t). Using (7.1)it is easy to check that Il fxllAm < Clxl and 11 fillAcR)< Clxl. Let S be the Fourier-transform of p,; then - d o ) / = I<S,fx)l sup 1ylol l / f x l l ~ ( ~ ; c6u ) Cllxl sup Iflo(. R

R

Returning to the proof of Proposition 8, we have, if 1x1 Q 1, I p,(x) - p,(O)l < Cl 1x1 119 11. If 1x1 2 1 , a(x) and o ( x ) are equivalent; hence for all real x , Ip,(x) - y(0)l Q c2a (4 1ly11, IIvII = SUP I P ) I ~ I R --

If Proposition 8 is proved for y(x) - y(O), the result also follows for y(x). From now on, it will be assumed that, for some constant C , and

We define p, by yk(x) = p) (Okx)and a trigonometric sum P, by P, (Okx) = ( L [ y k ]()x ); L is defined by (7.9)-(7.12). The frequencies of P, belong to E as we have already observed in Section 5.3. If lxl 2 I ~ k ( ~ >=l IP (Okx)l < c2a ( O k x ) llyll Q c21x1 a ( e k ) lly11. If lxl Q l ~ l k ( ~ )= I Ip) (Okx)I Q CZa ( e k x )lldI < CZa ( O k ) IIvII. Since w in (7.12) decays rapidly at infinity, sup lwpkl Q C3a ( O k ) Ilp,\l R

and (7.12) gives Iyk(x) - (Ly,) (x)I < C31xl a(Ok) llplll, which, for 1x1 < 1 , does not exceed C3a (Okx)[IylI.Hence I(Lyk)(x)( < C5a(Okx)for 1x1 < 1. For all real x, (7.12) gives I(Lyk)( x ) ( < C4a (0,) liyII. A fortiori, l(Lyk)(x)l Q C,a (Bkx)IIyII for all real x. Finally IP,(x) - y(x)l - ~ and IP,(x)l < a(x) Ilyll, which proves Proposition 8.
226

PISOT NUMBERS A N D SPECTRAL SYNTHESIS

PISOT NUMBERS A N D SPECTRAL SYNTHESIS

227

6. Bochner's property for a Banach algebra

6.3. Dual forms of Bochner's property

6.1. Let A be a semi-simple self-adjoint regular Banach algebra with a unit. If E is the compact spectrum of A, A can be regarded, by Gelfand's representation, as a subalgebra of the algebra C(E) of all continuous complex valued functions on E. The norm off in A is greater than 11 f 11 m , the sup. norm over E. For each compact subset K of E, A(K) is the restriction algebra to K of all functions in A ; I\ f [IAcK) = inf llgll for those g in A which equalf on K. If K = E, A(E) is the whole algebra A.

Let A* be the dual space of A normed by the dual norm. Since A is regular and self-adjoint, the support of each S i n A* can be defined. From now on elements S of A* are called pseudomeasures and llSllpMis the dual norm. For a finite subset F of E, let I(F) be the ideal of all f in A vanishing on F. An element S of A* which is orthogonal to I(F) is a measure ,u carried by F.

6.2. DEFINITION 7: We say that Bochner's property holds for A if there exists an increasing sequence F k , k > 1, ofjinite subsets of E such that the following implication is true f

E

C(E) and lim Il f

<

+ co

imply

(7.17)

k++oo

Proof: Apply the metric form of the Hahn-Banach theorem. Another easy lemma will be needed:

This means, very curiously, that the property of belonging to A for a continuous function f : E -+ C can be verified by looking atf on finite sets. Example 1: Let E = [O, 11 and let A be the Banach algebra of all complex-valued continuously differentiable functions f on [0, 11; 11 f [ I A = sup If 1 sup If '1. Let B be the Banach algebra of all fsatisfying a

+

CO.11

co.11

Lipschitz condition 1 f ( x ) - f(y)l = sup If ( inf C.

+

LEMMA^: Let F be ajinite subset of E and B the set of all complex measures ,u carried by F whose norm in A* does not exceed one. Then,for each complex valued function f on F,

< C Ix

- yl on [0, 11; we put 11 fllB

LEMMA 7: Let X be a separable Banach space, X* the strong dual of X, Y a linear subspace of X* and C > 0 a constant. Let B be the set of a l l y in Y whose norm in X* does not exceed one. Then the following two properties of Y are equivalent: (a) llxll < C sup I(x, Y)I (ifc= 1, llxll = sup I(x, v ) l ) ; YEB

YEB

(b) for each x* in X* there exists a sequence y,, k 2 1, o f elements of Y such that llykll < Cl(x"I)and, for every x in X,

co.13

Then for each finite subset F of [0, 11 and each f : F -+ C we have Il f IlacF) = Il f IIBcF). Since A and B are distinct, Bochner's property cannot hold for A. Example 2: If E = [O, 11 and A is the Banach algebra of all restrictions to E of 2n-periodic absolutely convergent Fourier series, then Bochner's property holds for A, as Theorem I11 shows. Example 3 : More generally let G be any metrizable compact abelian group and let A(G) be the Banach algebra of all continuous functions f : G -+ C whose Fourier series are absolutely convergent. Then Bochner's property holds for A(G). In the same context, Bochner's property holds for the algebra A,(G), 1 < p < co defined by Herz [56, 571.

+

An immediate consequence of Lemmas 6 and 7 is

PROPOSITION 9 :Let A be aregular self-adjoint semi-simple Banach algebra with a unit, and let E be the spectrum of A. Then A has Bochner'sproperty ifand only if there exists a sequence (Fk)k21ofjinite subsets of E such that for each f in C(E), the following two properties are equivalent: for each k Z 1 and each complex measure ,uk carried by F,,

I.f

d ~ k l

<

I l ~ k l l ~ *

(7.20)

228

PISOT NUMBERS A N D SPECTRAL SYNTHESIS

PISOT NUMBERS A N D SPECTRAL SYNTHESIS

In this case, for each S i n A* there exists a sequence ( I ~ k ) k 2 of measures such that p, is carried by F, (7.22)

7.2. A compact group G

Z. 8 + I ! I - ~ - " + ~Since For each k 2 1 , l e t r , be the subgroup 8-kZ + is an algebraic integer, r, c . We define the subgroup l' of R to be the union of all k 2 1 ; equally, we can say that T is the inductive limit o f r k ,k 2 I . We consider r w i t h the discrete topology and let G be the compact dual group. Sinceris countable, G is metrizable. Let j :I' + R be the canonical injection and h : R + G the dual map. Sinceris dense in R, h is injective; since j is injective, h(R) is dense in G.

rk+,

rk,

For every f i n A, ( p k ,f )

-+

(S, f ) as k -+

+ oo .

(7.24)

7. Bochner's property and harmonic analysis

7.1. THEOREM VIII: Let 8 > 1 be a Pisot number, N 2 1 an integer and E = E (N, 8) the compact set of all sums ~ ~,, E8, =- 0,1, . .., N. For each k 2

xk, 1 , let F, be thefinite subset of E consisting of the ( N + I ) , sums

Ckl F ~ O - ~E~, = 0, 1 , ..., N.

Then Bochner's property holdsfor E and F , , k 2 1. More precisely f

E

C(E) and

sup Ilf

/IA(F,)

<

+ co

implies

(7.25)

k21

229

PROPOSITION 10: Let P(X) = Xn + a, Xn- + . .- + a,, aj E Z for 1 d j < n, be the minimal polynomial of 8. Then the compact metrizable abelian group G can be regarded as a closed subgroup of the product group T ~G;consists of all x = (exp 2nip,, ...,exp 2niyk, ...) in T~ such that, for all k 2 1 , g,, alqk+, + + a,cpk+,=O (modl), a n d h : R-t G is then defined by h(t) = (exp 2niO-'t, ..., exp 2niO-,t, .. .). Proof: If x E Hom (T,T ) , let us put x(8-,) = exp 2nig,,, k 2 1. For eachk 2 1, we have 8-, + + + U , , O - ~ - ~= 0. Hence

+

+ a ~ g , k ++ ~ + a,g,,+, = 0 (mod 1 ) and the character x is well defined on r by the sequence of g,,, g,k

The compact set E is a set of synthesis and each distribution S carried by E whose Fourier-transform is bounded is the weak star limit of a sequence S,, k 2 1 , of measures carried by F, and such that llSkllpM < IIS(lpM. For each bounded continuous g, :R + C whose spectrum lies in E, there exists a sequence g,,, k 2 1 , of trigonometric sums such that (a) for each k 2 1, the frequencies of g,, belong to F,; (b) sup 1 ~ k l< sup 191; R

R

(c) g,,(x) y(x) uniformly on each compact set of real numbers. If 8 = 2 and N = 1 , Eis the interval [0, 11 and we get the original form of Bochner's theorem. The last part of Theorem VIII is a consequence of Bochner's property and of Theorem IV. The proof of Theorem VIII is a generalization of an argument given by Y.Katznelson when 8 = 3. The idea is to define a compact metrizable abelian group G which reflects the algebraic properties of 8 and to consider R as a dense subgroup of G. -+

k 2 1. Conversely, let (Q),),~ be a sequence of real numbers satisfying each congruence (7.27) and let us try to define a character x on T such that, for each k 2 1, ~ ( 8 - &=) exp 2niqk. Let xk E Hom ( r k ,T ) be defined by xk(@")

=

exp 2nivj, k

<j < k + n - 1 .

Since 8-,, ..., 8-k-n+1is a basis over Z of the free Z-module r k , this definition is consistent. It will be proved that ~ k 1+ Irk = X k and we can then define x E Horn ( r , T ) by XIrk = x,. To get (7.28) we can restrict our attention to the generators 8-j, k < j < n - 1, of the free Z-module 8-k is written in rk+, as a sum -ale-k-l - ... - a,8and then ~ ~ (,) 8=- exp 2niq1, while xk+ (O-,) = exp -2ni ( a , g ~ ~++... ~ + a,gJk+,). The required equality is implied by the congruence (7.27). For k + 1 < j < k + n - 1 , each 8-j in is written 8-j in and xk+,(8-j) = xk(8-j).

,-"

rk

rk+,

rk;

230

PISOT NUMBERS A N D SPECTRAL SYNTHESIS

PISOT NUMBERS A N D SPECTRAL SYNTHESIS

If x : R T is a continuous character defined by ~ ( x=) exp 2nixt ( t E R ) , h(t) is the restricted character xlr E Hom (T, T ) . Clearly h(t) = (exp 2ni8-kt)k2 . -+

7.3. Extension to G of Fourier transforms of measures supported by F, It may be noted that each F,, k 2 1 , is contained in r a n d so therefore is the union U F, of all the F,. For any measure p carried by F,, 6 : R -+ C k21

is a finite trigonometric sum whose frequencies belong to (to be more precise, the frequencies of 117, belong to -F,). Hence there is a unique finite trigonometric sum P : G -+ C whose frequencies and Fourier coefficients are those of 117, such that p = Poh. (7.29) Since h(R) is dense in G, we have

LEMMA 9 : For each jixed p lim M,,,(x) = 0.

> 1 and each x E G,

= G\h(R),

q++m

If the limit of the decreasing sequence IM,,,(x)l, q > 1, is ~ o 0t, lcos nrp,.,l -+ 1 and rp, = p, + r,, p, E Z , -5 < r, < +, lim r , = 0. k++m

+

But for each k 2 1, we have q~, a1qk+,+ + anqk+,E 0 (mod 1). a,r,+,, we get t , E Z and lim t , = 0. + Putting tk = r, + a,rk+l + This implies t , = 0 for k 2 k,. Let 8,, ..., 8, be the conjugates (over Q) of 8. The linear recurrence relations r , + a1rk+, + + a,rk+, = 0 ( k 2 k,) imply r, = tovk + t 2 8 i k + ... + t n 8 i kfor k k,, a suitable real t and suitable complex numbers t,, ..., t,. By hypothesis 8 is a Pisot number: 1 8,l < 1, . .., l8,l < 1. Since r, tends to 0, the only possibility is r, = to-, (Chapter I , Section 6.3) or rp, EE (mod 1) f o r k 2 k,. But, for each k 2 0, O V k + a18-k-1 ... a,8-k-n = 0 and rp, + a,rp,+, + . + a,rpk+, = 0 (mod 1). By a decreasing induction on k 2 0 , we get rp, = t o - , (mod 1 ) for each k 2 0 and so x belongs to h(R). 0

.

.

+

+

We denote by d the vector space of all P : G -+ C defined by (7.29); d is normed by (7.30).

LEMMA 8: Let p 2 1, q 2 1 be two integers and ,up,, the probability measure carried by 8-pFq and dejined by the convolution product

7.4. Conclusion of the proof of Theorem VIZl Let C

=

lim

1l.f \ I A cFk,

and let L be the linear form defined on B by the

k++m

where vj is the sum of a mass 112 put in 0 and of a mass 112 put in 8-j. For any continuous f : E -+ C, lirn J E f d , , , , = f(0) uniformly in P+ + m q 2 1. Proof: O-pFF,c 8-'E tends to 0 a s p tends to infinity; hence ,up -+ d(x) in the weak-star topology a ( M ( E ) , C(E)),uniformly on q 2 1. Roughly speaking, the following lemma states that, using a suitable sequence of elements of b,we can separate G, = h(R) from its complement G, in G. For each real t , we have

$,,,(t)

=

2-4 ( 1

+ exp -2nitO-"-l)

... ( 1

+ exp - 2 n i t 1 3 - ~ - ~ ) .

The continuous extension M,,, of P,,, to G is defined by

Mp,,(x)

=

2-4 ( 1

whence IMP,, (x)l

=

+ exp -2niq,+ ,) .-.( 1 + exp -2nipP+,)

lcos np,+ ,I

... 1 cos nq,+,I.

following rule: if p is a complex measure carried by one of the F, and if the continuous function P : G -+ C is determined by

Poh

=p,

then L(P)

=

Proposition 9 shows that IJFkfdpl < 1l.f

IFF.

sup R

(7.32)

< C sup IPI. G

Hence L is continuous and the norm of L does not exceed C. By the metric form of the Hahn-Banach theorem, L can be extended to the whole of C(G) without changing its norm: hence there is a complex Radon measure o, lluli < C, such that L(P) = JG P do. Therefore (7.32) can be rewritten

Jif dP

= JG

p do

for any complex measure ,u carried by one of the F , , where P : G the continuous function defined by P 0 h = $.

-+

C is

232

PISOT NUMBERS A N D SPECTRAL SYNTHESIS

PISOT NUMBERS A N D SPECTRAL SYNTHESIS

The real line is the union of an increasing sequence of compact sets; since h is continuous, h(R) = G1 is a Borel subset of G and so is G, = G\Gl . Let o1 be the restriction of o to G, and o , the restriction of o to G, . If it can be shown that o, = 0, the proof ends in a few lines. Under the action of h, the Borel field of R is carried onto the restriction to G I of the Borel field of G. Hence each complex bounded Radon measure ol on GI' can be written o1 = h(z) where 7 is a complex bounded Radon measure on R and [lo,I [ = [[TI[. This implies jR f d = jG P do = jG P do, = jR@ dz = jR Z dp. The equality between the first and the last term implies f = Z on each F,, k >, 1. Since the union of all the F , is dense in E, f = Z on E. Hence [[ f [lB(E) < C. But E is compact; A(E) and B(E) are isometric (Chapter IV, Proposition 5) and Theorem VIII is proved.

7.5. o2 = 0. We return to the definition of the number C

=

lim k+

[I f

+ oo

[/a(F,)andwe apply Proposition 9 : for each positive E,there is a k 2 1 and measurep carried by Fk such that sup l@l = I and (jF,fdpl > (1 - s) C. R

+

For each q 1 and each p > k, F, I ~ - ~ cF Fp+, , and Lemma 8 shows that for sufficiently large p and each q > 1

(p,,, is defined by (7.31)). From now on p is fixed. Since p *,up, is carried by F,+,, our fundamental relation (7.33) can be applied to get

IJ.

P(x) Mp,,(x) do (x)

8.1. Before giving any precise results, some definitions should be recalled. Let Q , be the p-adic field and A, the subring of all mpn in Q , ( m E Z , n E Z and m andp are relatively prime). Thep-adic norm is defined on A, by Impn[, = p-" and A, is dense in Q , for this norm. A Pisot-Chabauty p-adic number 8 is defined by the following four properties: (a) [el, > 1; (b) 8 is an algebraic integer over the ring A,; let n be the degree of 8 and let P ( X ) = Xn + clXn-I + .-.+ cn (cj E A, for 1 < j < n) be the minimal polynomial of 8; (c) if 4, is the algebraic closure of Q,, all the other roots z of P in 4, satisfy [zl, < 1; (d) let z,, . . .,zn be the complex roots of P ; then It,[< 1, .. , lz,[ < 1.

8.2. THEOREM IX: Let 8 be a Pisot-Chabauty number and E(8) the compact set of allp-adic numbers x = r F sk8- k,E, = 0 or 1 Then E(8) is aset of synthesis. Moreover for each bounded continuous g,: Q , + C whose spectrum lies in E(8), a sequence (vk),> of trigonometricsums can befound with the following properties: k

~ ~ 8 -E~j ,= 0 or 1

the frequencies of y k arefinite sums

(7.34)

1

> (1 - 2s) C ,

where @ = P 0 h and A ,, = Mp,, o h. Writing jG PM,,, do = j,, PM,,, do, ,.j PM,,, do, = I J, we observe that [ I ]< I[ol[l,since sup IPl = sup l@l = 1 and sup IMP,,[ = 1,

+

G

8. The p-adic case

R

+

G

acd that I J I 6 jcj?IMp,q[ db21. Hence, (1 - 2 ~ C) < llol[[+ j~~ IMP,,[ dlo2[.Let p be fixed acd let q tend to infinity. Lebesgue's dominated convergence theorem and Lemma 9 show that the last integral tends to 0. Hence (1 - 2s) C 6 [lol[I for each positive E , which implies C < Ilol 11. But llolll + [[021[ = [lo[[ 6 C ; we get [lo2[[< 0 and so o2 = 0. Theorem VIII is now completely proved.

g,,

+ g, uniformly

on each compact subset of Q,

(7.36)

8.3. As in the real case, the proof of Theorem IX can be divided into two parts. First of all, a weak version of Theorem IX is given where g,, depends linearly on g, and where (7.35) is altered; this is the following proposition : PROPOSITION 11 : Let 8 be a Pisot-Chabauty number. There exist afinite subset F of E(8), a positive constant C and a sequence Lk,k > 1, of linear maps such that, for each bounded continuous g, : Q , + C whose spectrum lies in E(8),

234

PISOT NUMBERS AND SPECTRAL SYNTHESIS

23 5

PISOT NUMBERS AND SPECTRAL SYNTHESIS

the frequencies of the trigonometric sum Lk(g7) are finite sums

Let P(X) E Z [XI be the minimal polynomial of 8: P(X) alXn-l + + an, where a, = pr = 181, (r 2 I).'

=

aoXn

+

m . 0

PROPOSITION 12: Regarded as a subgroup of TN, G is the group of all sequences o = (ok)k>lof complex numbers of absolute value 1 such that, for each k 2 1 and for real q~, ok= exp 2nicpk (7.41) and aoyk anvk+,= 0 (mod 1). (7.42)

+

+

With this representation of G, the duality between G a n d r is given by the in G pk8- in r and each o = following rule: for each y =

xk,

Proposition 11 is an easy consequence of Theorem XI of Chapter IV, and can be proved exactly as in the real case.

xv(o) = exp 2ni

pkcok. k> 1

8.4. The second state in the proof of Theorem IX is the following result, where E = E(8) is d e h e d as in Theorem IX.

r;

THEOREM X: Let f : E -t C be a continuous function and C a positive constant. The following two properties off are equivalent: (a) for any complex measure p carried by ajinite subset of E

(b) f is the restriction to E of theFourier-transform of afunction in L1(Q,) and Ilf IlAcE) < C. The implication (b) => (a) is obvious. The proof of (a) => (b) is very similar to that of Section 7. The idea is to construct a compact group on which& can be extended by continuity and to apply the Riesz representation theorem to describe the linear form P dp. +

I

LEMMA 10: Let m 2 1 be an integer. For each positive E and each sequence ( P ) ~ ksm ) ~ of numbers in A, satisfying (7.42), there exist an x in A, and m p-adic integers pk, 1 < k < m such that x8- + p k belongs to A, and

SE~

LEMMA 11 :Let I >, 1 be an integer and let w be an element of TNsatisfying (7.41) and (7.42) and such that ?, = 0 for I 1 < k < 1 f n. Then there is an x in p-"-'Z such that (7.45) x O - ~ = g l k ( m o d l ) for 1 < k < 1 n.

8.5. Definition of G

r

I,,

+

+ ..- + rk.

For each k 2 1, let rk be the group Z8-I Z8-k (without topology) and let be the inductive limit of the We regard r as a discrete group and denote its dual by G. Our discrete group r is a quotient of the direct sum of a countable set of copies of Z : to each finite sequence p = (p,),,, of rational integers corresponds pkrYkin Hence G is a subgroup of the compact product group TN of a countable set of copies of T.

Proof of Proposition 12: Let G' be the subgroup of TNdefined by (7.41) and (7.42). We describe an injective mapping i : G + G'. Each element x of G is a character on putting i(x) = (ok)k,l where ok= ~ ( 8 - we ~) get (7.42) from + ... + an8-k-n = 0 (k 2 I). Clearly i is continuous. = prZp. Let j* :r+ f' be the Let f' be the closure of F in Q,: identity mapping. Then j* is the dual of j: Ap/prZ+ G and i o j ( x ) = (exp 2 n i ~ € J - ~for ) ~ each , ~ x in A, (mod. prZ). To prove the surjectivity of i, from which it will follow that i is a homeomorphism, it suffices to show that I: A, -t G' defined by I(x) = (exp 2 n i ~ O - ~ ) ~has , a dense range. The following lemmas conclude the proof:

+ .

r.

I

i

. a = pr, r > 0, since P is minimal and all a&, E Ag.Let O, . .,6, be the other roots of P in LIP.Then 10lp > 1 and IOzlPG 1, .. ., 10nlpG 1 imply leip = 10 0, = lal/aolp 6 pr, If 101, < pr, the other symmetric relations between the roots of P give laJ/aolp< pr for 1 G j 6 n; p divides ao,.. ,a, and P is not minimal. Here It1 denotes the absolute value of the real number t E A,.

.

+ + +&Ip

PLSOT NUMBERS A N D SPECTRAL SYNTHESIS

PISOT NUMBERS A N D SPECTRAL SYNTHESIS

236

We begin with the proof of Lemma 11. Let GI be the group of all ~ E T ' + " defined by (7.42) such that p, = 0 for k 2 I + 1. Computing successively cp,, ql-, , ... we get pr possible choices for pl (mod 1) and for each of these choices, pr choices for 9,-, and so on. Hence Card GI = p". Let h : p-lr-rZp/p-rZp + G1 be defined by h(x) = (xO-~)~,,, where each term is taken modulo 1. Then, the kernel of h is equal to (0) and since Card (p-('+'" Z,/p-'Z,) = p", h is surjective. Since x is defined modulo p-'Z,, x may be chosen in p-(l 'Ir Z. To prove Lemma 10, it is convenient to consider first the case m = n. Let h : A, + Tn be the homomorphism defined by h(x) = (exp 2nixO-l, ...,exp 2nixO-"); then h(Ap) is dense in Tn. If this was not the case, a sequence p, , . ..,p, of rational integers could be found such that for each x in A,, x(p18-l + -.. + pn8-") G 0 (mod Z,). The only possibility is p18-l + + pn8-" = 0 ; as the degree of 8 is n, we getp, = = pn = 0. Hence no proper closed subgroup of Tn contains h(A,). If m 2 n + 1, m = n 1 say. We approximate (ql+l, ..., TI+,) by modulo ( x ~ e - ,~..-.,~x18-'-"). Then the difference (p, - x, 8-,), ,,, (Zp)m,is very close to an element of GI, which can be written as a sequence ( ~ ~ e - ~ ) ~ < modulo k < m , (Z,)", and x = x, + x, gives the requiredelement of A,. +

-

0

-

+

8.6. We return to the proof of Theorem X. For each j 2 1, let vj be the measure assigning the mass 3 to 0 and the mass 3 to 8-j, and for all j > 1 and k 2 0 let Pj,k = v j + l * " ' * vj+k. As in the real case, MjSkis the extension of ,4j,kto the whole of G and we have I+,

LEMMA 12: Iffor some fixed j, lim MI,,(@) f 0, then there is an x in k++m A, such that = ( ~ 8 ,)k2 - 1. Proof: If this is the case, lcos nqkl + 1 and 9, = p, + r,, where p, E Z and r, + 0. The congruences (7.42) give the corresponding ones for r, (when k > m). Let t , , ...,tnbe the complex roots of P(X) = 0. Then, + ..- + 2,~;~. Since rk + 0 as k + f a,we get for k > m, r, = 2, = ... = A n = Oandr, = O f o r k > m. Lemma 11 now gives the result. The last part of the proof is now a straightforward transcription of that given for the real case.

23 7

9. Notes

An excellent survey of the problem of synthesis for Banach algebras can be found in [14] (where sets of synthesis are called Wiener-sets). For 8 = 3, Theorem V was proved by Herz in [45]. The general form of Theorem V was conjectured by Salem.

ULTRA-THIN SYMMETRIC SETS

CHAPTER VIII

ULTRA-THIN S Y M M E T R I C SETS

We define a class of symmetric compact sets E of real numbers which are very similar to Cantor sets constructed with a dissection ratio whose inverse is a Pisot number. Such an E will be a set of uniqueness and of synthesis and all restriction algebras A(E) will be isomorphic; these E will be called ultra-thin symmetric sets. Other group algebras, like A(Dm), can be regarded as subalgebras of A(E) and, following Varopoulos and Katznelson [93, 111, this identification can be used to obtain a compact subset K of E which is not a set of synthesis.

1. Properties of ultra-thin symmetric sets We recall the definition of symmetric sets of real numbers (already given in Chapter 111, Section 2): we start with a compact interval [a, b] of real numbers and a sequence (t,),,, of real numbers belonging to the open interval (0, +); E is the ictersection n E, where E, is a union of kg1

2, intervals of length t , 5, ( b - a ) and is defined inductively by the following rule : to construct all the intervals of E,, each compact interval I of Ek-, is replaced by two compact intervals I' and I " of length tklI1 ;the left-hand end point of I' is that of I and the right-hand end point of I " is that of I; E, is [a, b]. E is called ultra-thin $27 5; < +co. As usual, the Banach algebra A(R) is defined by the rule: f E A(R) iff = Ilgll For each is the Fourier-transform of g E L1(R) and Il f compact set E of real numbers, A(E) is the Banach algebra of all complex valued continuous functions h defined on E which are the restrictions to E of functions f in A(R); IlhllAcE,= inf Il f llAcR, over suchf. If E is contained in the open interval (-n, n), and this is assumed in everything that follows, A(E) is also the Banach algebra of all restrictions to E of absolutely convergent Fourier seriesz?: ckeikXwhere C?: Ic,l < + co ; these two definitions of A(E) lead to equivalent norms. All ultra-thin symmetric sets are not affine copies of each other, nor

239

even locally affine copies. However, within the scope of harmonic analysis, all ultra-thin symmetric sets are the same as the following result shows: THEOREM I : All ultra- thin symmetric set^ E are sets of uniqueness and of synthesis and all the corresponding restriction algebras A(E) are isomorphic. The role played by the exponent 2 is crucial since, for each p > 2, we shall construct, following Salem, a sequence (tk)k,l of dissection ratios such that zy 5; < + co and such that the corresponding symmetric set F has the two properties: F is a set of multiplicity and for each ultra-thin symmetric set E, A(E) and A(F) are not isomorphic.

2. Functions whose spectra lie in ultra-thin symmetric sets 2.1. To get a more precise version of Theorem I, it is helpful to give another definition of ultra-thin symmetric sets. DEFINITION 1 : Let (tk)ka be a sequence of positive real numbers satisfying the two conditions:

for each k 2 1, tk > s , + ~= t k f l + tk+2 +

(Y)~

(8.1)

< +m.

and k, 1

The corresponding ultra-thin symmetric set E is the compact totally disconnected set E of all sums x = ZF cktk,E, = 0 or 1. 5k-l (1 - tk), a = 0, b = 1, we see that this new Putting t , = El definition is equivalent to the first one. DEFINITION 2: With the notation of De$nition 1, we denote by (a) F, thefinite set of the 2, sums cjtJ,E~ = 0 or 1; (b) E, the set of the 2, intervals whose origins belong to F, and each of whose lengths is s, + ,; (c) I , the interval [0,sk+J. We then have, Fk c E c F, I, for each k 2 1. We shall now describe the space of all complex valued bounded continuous functions y whose spectra lie in E.

x:

+

3 : For each k 2 1, we write p, as a perturbed trigonometric DEFINITION sum

ULTRA-THIN SYMMETRIC SETS

ULTRA-THIN SYMMETRIC SETS

240

+

where the spectrum of each g?, lies in I,, g?, = g,(Ei,...,Ek) if jl = t k t kand E , = 0 or 1, the second sum being extended over the 2, such sequences of 0's and 1's. For all real numbers x,xl , .. ., x , and with the same summation convention, we put

+

24 1

sake of simplicity, it will be assumed that

where we have v2 = 1 IT2. Now g,(t) becomes an almost periodic function with nine coefficients; the frequencies off are clvl E ~ V Zfor E , ,E~ = - 1,0 or 1. Better observations show that each of these nine coefficients off itself has 'secular perturbations' whose period T , is very large compared with T 2 and so on. If g, is assumed to have a 'nice' behaviour at infinity, for instance if g, is bounded, a mathematical model can be built to explain the 'projective sequence' of approximations of g,, to compute the corresponding error terms and to find the global behaviour of g,. Let E be the set of all infinite sums ekvk,E , = - 1 , O or 1. (We assume that the sequence T,, k 2 1 , of periods of 'secular perturbations' grows sufficiently rapidly to satisfy l / T k < + co.) Two cases have been studied in this book: (1) We have z y (v,, 1/vk)2< + co. Then this mysterious function g, is any complex-valued bounded continuous function on the line whose spectrum lies in E. For each k 3 1, we define the finite set to F, consist of the 3, sums &,vl + .- E,v,, E , = - 1,0 or 1. Then, for each k 2 1 , d t ) is identically zAEFkg,,(t)exp 2nijlt, where the spectrum of each g,, lies in [0,+,! ,I, where +,!, =vkfl V,+Z + -... We put g,, ( t o , t ) = C A E F k g,,&) exp 2niAt and -. we get Ig,(t) - g,, ( t o ,t)l < C (It - toll TT:~ sup IyI. Hence if L , = JT,T,+, and to = 0 , Ig,(t) - g,, (0, t)l <

+

and

P)L(X) =

~ ( 0exp ) 2niilx =

Gk(0, t,x, ..., t,x).

2.2. With this notation we have 11: For each ultrathin symmetric set E of real numbers, there THEOREM exists a constant C such that each complex valued bounded continuous function g, whose spectrum lies in E can be approximated in the following way: 6) for all real x, Ipl(x) - g,,(x)l < Ctk, 11x1 sup Id; (ii)

SUP l p l k l R

<

SUP Rk+l

1&1 < C SUP

R

lyl;

R

(iii) on each interval I of length C/t, , sup lplkl R

< C sup I F , ]

.

I

Roughly speaking (i) describes the quality of the approximation of the perturbed trigonometric series g, by the corresponding uniformly bounded pure trigonometric sums g,, and (iii) shows that there exists a kind of periodicity of period Clt, for these pl,. Finally (ii) shows that the frequencies t , are decreasing so rapidly that sup-normscan be estimated byreplacing each exp 2nitkx by an independent exp 2nixk. The proof of Thecrem I1 will be given in Section 4. The intuitive content of Theorem I will be described now. A problem raised in the introduction should be recalled. A function of time is denoted by g,(t) ('function' here means complex valued continuous function). Observations are made over a long interval [0, L,]. The first feeling is that g,(t) is periodic. Let T l be this period and v, = T;' the corresponding frequency. For the sake of simplicity, it will be assumed that g,(t) = a(0) a ( l ) cos 2nvlt a(2) sin 2nv2t. But more careful observations over longer intervals show that a(O), a(1) and a(2) are actually three functions of time whose period T2 could not be detected on the interval [0,L1]; T 2 is very large compared with L, or with T I . For the

+

+

Cy

z:

- +

+

R -

C JT,/T,+, sup ITI on [0,L,] and we cannot distinguish between g,(t)and R

(09 t ) = g,kW A possibility exists for a local harmonic analysis of g, on [0,L,]: the observation of g,(t) = zrlEFkpA(t) exp 2nik on [0,L,] givesestimates of each perturbed coefficient ~ , ( t )on [0, L k ] : an approxin~ationof q,(t0) is ( l / L k ) ~ : ~ + L k g , ( t )-2niAt e x p dt ar,d the error term is bounded by

plk

C' J T k / T k + SUP I T I . R

The global behaviour of g, can be very different from the expected behaviour of an almost periodic function. For example, let ,u be a continuous measure carried by E and let ~ ( t=) @(-t). Then a theorem of Wiener ([24],Ch. XVI, th. 4.19) shows that lirn T-I JiIg,(t)l dt = 0. T*+m

242

ULTRA-THIN SYMMETRIC SETS

Since the derivative rp' of rp is bounded, this property implies that for each positive E and each integer n 2 1, there exists an interval of real numbers of length n on which lrp( < E. (2) 0 is a Pisot number, p is a large integer and Tk/Tk+, = vk+,/vk = 0-". Here also, our mysterious function rp is any complex-valued bounded continuous function on the line whose spectrum lies in E. All the preceding analysis of such a rp remains valid; C and C' are now two constants depending only on 0.

ULTRA-THIN SYMMETRIC SETS

243

Let rp be as in Theorem I1 and let f be a function in L1(R), whose Fouriertransform vanishes on E. Then, Lebesgue's dominated convergence theorem together with (i) and (ii) gives lim JR y,(-x) f(x)dx k*+, = JR rp(-x)f(x) dx. But, by an easy computation, JR plk(-X) f(x) dx = x l E ~ k r p A ( 0 ) f= ( ~0.) Hence JRrp(-x) f(x) dx = 0.

To prove this remark, it helps to regard E as a compact subset of (-n, n); since E i s a set of synthesis, the dual space PM (E) of A(E) is the Banach space of all distributions S carried by E such that sup ~L?(n)l R < +a. If A, is not dense in A(E), there exists an S # 0 in PM (E) such that (S, f ) = 0 for each f in A,. Iff = 1, this gives $0) = 0 and integrating the Fourier series s(n) e'"" of S, we obtain the Fourier series C?: $(n) einx/in of a function F in L2 (- co, + co); the distributional derivative of F is S. Hence F is constant on each interval of the complement of E in [-n, n] (since S is carried by E). For each x and x' in (-n,n)\E the restriction to E of the characteristic function of [x, x'] belongs to A,. Hence F(xr) = F(x) and F is constant on the complement of E ; since E has Lebesgue measure 0, F = C, where this equality holds in the sense of distributions, and S = 0. We return now to the proof of the isomorphism. Let E' be another ultra-thin symmetric set defined by a sequence satisfying (8.2) and let H: E' + E be the homeomorphism defined by H ~ , t @= cktk for all E, = 0 or 1.

3.2. Each ultra-thin symmetric set is a set of uniqueness

PROPOSITION 1 : For each f E A(E), f o H belongs to A(Er) andfor some constant C depending only on E and E'

3. Theorem I as a corollary of Theorem I1

3.1. Each ultra-thin symmetric set E is a set of synthesis

Let rp : R + C be a complex-valued continuous function vanishing at infinity whose spectrum lies in E. Since t,+ ,It, + 0 and t, + 0, there exists an increasing sequence (x,),,, of positive real numbers such that t,x, + + co, while tk+,xk+ 0. Let I, be the interval [x,, x, + Clt,] and let w , = sup (pi; by the assumption on q~,w , + 0 as k + + CQ. Ik

SUP Irpkl

< Ctk+l (xk + C/tk) llrpllm + w k which

tends to 0. (iii) now

Ik

shows that sup (p,l also tends to 0 and so the limit rp of rp, is identically 0. R

3.3. All Banach algebras A(E) are isomorphic Befcre beginning the proof, we describe a dense subalgebra of A(E). For each k 2 1, let A, be the subspace of all f in A(E), which are constant on each interval 1 I,, 1E Fk(Definition 2) and let A, be the union of all the A,. Then A , is dense in A(E).

+

x,

2:

(1''

This proposition gives the required isomorphism since interchanging E' and E yields the reverse inequality. In proving (8.6) we may assume that f belongs to A,; the general case then follows from the density of A, in A(E). To compute norms in A(Er), we test on the unit ball of the dual PM (E') : let rp' be any complex valued continuous function whose spectrum lies in E' and such that llyfll,< 1, let Sfbe the Fourier transform of rp', let pl; be dcfined in terms of pl' by (8.5) and let S L be the Fourier transform of 9;; S L is a measure carried by FL. I f f ~ A , , f o H ~ A L a r d ( f o H , S '=) (foH,SL) =
2

exp 2ni (clxl 4-

-.-+

< Ilf I ~ A ~ E ,

< Cllf I I A ~ E ) .

IIplkll m

Rk H, S')I =

Ix

E,x,)~ < C. Then I( f 0 ( f ,H(S;)> Taking the sup over Y', We get (8.6)

v ULTRA-THIN SYMMETRIC SETS

ULTRA-THIN SYMMETRIC SETS

Using exactly the same argument, the following result (used in Section 7) can be obtained.

LEMMA 1 : Let q > 1 , w > 0, 0 < I < llqw, let I be the interval [0, I ] and let h : R + T be the continuous homomorphism dejined by h(t) = exp 2niwt. There is a constant C depending only on q such that,for each pseudo-jiunctio~zS carried by I

244

2: For each ultra-thin symmetric set E of real numbers PROPOSITION there exists a constant C such that, for each sequence 81, ..., 8, of n real numbers, the function f : E -t C defined by

belongs to A(E) with a norm not exceeding C.

245

Roughly speaking, Lemma 1 asserts that the local isomorphism between R a c d T defines almost an isometry for pseudo-measure norms of distributions carried by a small interval. Proof of Lemma 1: Let S be a pseudo-function carried by I and let y(x) = S(-x). We have to prove that sup lyl < ( 1 + C12w2)SUP IqI; let R

mZ

4. Proof of Theorem 11: reduction to a problem on intervals

x , be a point where Iq(xo)l = sup 1471: such a point exists since q is

4.1. THEOREM 111: Let or, t l , ..., t, be an increasing sequence of n + 1 positive real numbers and let q > 1 be such that t1 > qol, t2 > q ( t , + a), ..., t , 2 q ( t , - l ... t , a); C will denote a constant dependingonly on q. Let A , be the set of the 2" sumsx: sktk,ck = 0 or 1 and let a,(x), ilELI,, be a sequence indexed by A, of complex valued bounded continuous functions whose spectra lie in [0,or]. We put

continuous and vafiishes at infinity Then. by Taylor's formula, q ( x ) = y(x,) ( x - x,) A 2-' ( x - x,)' B(x), where A = yt(x0), 1B1 < sup IqV(x)l.By Bernstein's inequality (Chapter V, Section 4 ) sup Iq"l

+

+ +

R

+

+

R

R

< 4n2I2sup lql = 4n212 Iq(x0)l. Sicce Iq~(x,)lis the maximum of all Iq(x)l, R

the minimum of ly(xo) + ( x - x,) A1 is obtained when x = x , and A/y(xo) is a pure imaginary (see Chapter I, Section 4.3, Lemma 7). We get (y(x)I 2 Iy(xo)l (I - 2n2121x - xoI2). There is an x G o Z such that Ix - x,l < w/2 which implies sup lql 2 (1 - n21202)/2)sup 1yl.l aZ

R

LEMMA 2 : Let q > 1 , 0 < I < ql < v, and let a,(x) and al(x) be two functions vanishing at infinity whose spectra lie in [0, I]. Then suplfl R

<

sup

R"+1

1.71< ( 1

sup la,(x)

+ co12/t:).-(1 + Ct,Z-l/t,Z)SUPIfl. R

4.2. Two lemmas: high frequencies and independence Let U be the union of all intervals [A,1 + or], 1 €A,, and let PF ( U )be the set of all distributions S carried by U whose Fourier-transforms vanish at infinity; we put IlSll = sup lS(x)l. xsR

Then, as we observed in Chapter IV, Section 14, the dual of PF ( U ) is A(U), whose dual is PM (U). Using this fact, the proof of (8.10) can be reduced to the case where f tends to 0 at infinity. The following two lemmas will be needed:

+ eznivxal( x ) ]< sup lao(x) + ezxiYal(x)l

R

R2

Roughly speaking, the effect of the high frequency v is to make e2""" independent of the low-frequency terms a,(x) and al(x) (this observation has a very concrete interpretation: the position of the second hand of a watch gives very little information about the position of the hour h a ~ d ) . Proof: Let 0 be a fixed real number ar.d p ~x t= ( I 3 + k)/v,k E Z. We have sup lao(x) ez"ivxal( x ) ]> sup la, ( B Y - l e2""a, (8v-l kv-')l.

+

+

R

+

ksZ

If Iw E [n-l,q - I ] . the proof of Theorem 11, Chapter 111, shows that IISIIPM(R) G C, ~ ~ ~ ( S ) I I P M ( T ) .

246

ULTRA-THIN SYMMETRIC SETS

ULTRA-THIN SYMMETRIC SETS

Let b : R -, C be defined by b(t) = a, (0v-I + t ) + e2""a1 (Ov-l the spectrum of b lies in [0,I ] and Lemma 1 gives

+ t);

247

spectra lie in [0, I ] , SUP R

Finally, for each real 0, sup la,(x)

+ eZniBa1( x ) ]< ( 1 + C I2/v2)SUP

R

a,(x)

+ eZnvixa, (x)l. Taking the sup over 8 we get Lemma 2.

Then there exists a constant C2 depending only on C , such that for each pure c, exp 2ni1x, trigonometric sum P(x) =

R

and, assuming f is defined as above,

I

4.3. Proof of Theorem 111 We proceed by induction on n. Let s,-I = a + t , + .-.+ t,-, and for each function f whose spectrum lies in [0,a ] + A,, let f ( x , x , , . . ., x,) be the n + 1 dimensional extension off given by (8.4). We can write f ( x ) = g(x) + e2"'tnxh(x),where the spectrum of g and that of h both lie in [0, s,- ,I. The inequality s,- < ( q llq) t,-, and Lemma 2 give

,

sup I&) R2

+

+ eiYh(41< (1 + c1 t L / t 3 SUP If R

1,

(8.11)

SUP

R"

lg" ( x , x l , ..., x,-,)

. Hence

+ eiyh( x , x,, ..., x,-,)I

Combining (8.1 1 ) and (8.12) we get Theorem 111.

4.4. Proof of Theorem 11 Theorem I11 trivially gives (ii) in Theorem 11; the convergence of the t:+Jt; ensures the convergence of the infinite product series n k S l (1 + C tk2+11tk2>. To get (i) and (iii), we need only apply the following lemma:

xkS1

(8.15)

R

Proof of Lemma 3: Let A be the function which is 0 outside [0, 11, linear on [0, +] and [t,11 and such that A(+) = 2 ; A is the Fourier transform of y. Define C2 by the condition that Iy(x)l < 1/2C1if 1x1 > C, and apply (8.14) with a,(x) = y (Ix) a, for all 1 EA. We get C , sup lyPl 3 sup /PI. But sup lyPl R

R

<

sup

lyPl

1x19 1-1C2

+ 1/2C1 IIPll

R m,

which gives

(8.14). To get (8.15), we put f ( x ,y) = a,(y) exp 2niilx. Applying (8.13) to f ( x y), where y is a parameter, we get sup I f ( x , y)] < C1 SUP I f(x)l.

xAeA

+

where C1 = (q + l/q)' C and C is defined by Lemma 2. For each fixed y, the spectrum of g(x) + eiYh( x ) lies in A,-, the induction hypothesis gives

x a,(O) exp 2niAx - 6 a,(x) exp 2niAxl < 2nlCl 1x1 sup I f 1 .

&A

R2

R

Let us compute f ( x , y,) - f ( x , y,); for each fixed x, f ( x , y) is a function whose spectrum lies in [0, I]. By Bernstein's inequality, we get, for . we each fixed x , I f ( x , y 1 ) - f ( x , y 2 ) l < Iyl - Y Z I 2nl sup I f ( x , ~ ) l If yeR

replace y , by 0 , y2 by x and if we take the supremum with respect to x , we get (8.15).

5. Symmetric coherent sets of frequencies 5.1. THEOREM IV: Let (t,),,, be an increasing sequence of positive real numbers such that t,,, > t , + + t , = s, andx," (tk/tk+1)2< + CO. L e t A be the set of a l l j n i t e s ~ m s ~ ~ ~ E , = 0 or 1. ThenA isa coherent set of frequencies. Moreover for each positive E ,there exists a constant T ( E ) a, exp 2niAx, sup IPJ such that for each trigonometric sum P(x) = R <(I+&) sup IP(t)l.

xnen

It1 S T ( E )

LEMMA 3: Let A be a uniformly discrete set of real numbers: 0 < I < inf 11 - 1'1 for all 1 # 1' in A, and assume that for some constant C1 and each finite perturbed trigonometric sum a,(x) exp 2ni1x = f ( x ) , constructed with complex-valued bounded continuous functions a, whose

rneA

5.2. Before giving the proof,we make some remarksconcerning Lemma 3 : if, for some E > 0 , C1< 1 + &/2,we may choose C , = C2(&), in the notation of the proof of Lemma 3, sufficiently large to ensure Iy(x)l < ~ / 2

ULTRA-THIN SYMMETRIC SETS

ULTRA-THIN SYMMETRIC SETS

248

+ E if 1x1 > C 2 , which implies

by our choice of m. Applying (8.16) ar,d Theorem I11 to the set A of all finite sums ~ k , , , , ~ k t E~ k , = 0 or 1 and to I = dlrn,tm, we can findan x in [y, y + &/27csm- such that

C2(s)is icdepe~dentof I and A. Definition of an integer m: if Cis the constant of Theorem 111, an integer m can be found such that (1 Ct,- ,It,) ( 1 + ~t:/t;+,) . . (1 + ~ t z / t : +1 ) ' < 1 ~ 1 2 . This is possible sincex? (tk/tk+ < + a.Taking a larger m if necessary, we may also assume that

+

+

249

From Bernstein's inequality applied, for this fixed x, to the trigonometric sum t -+ Pn ( t , x , ..., x ) whose spectrum lies in [0,s,-,], we obtain IPn(y,x, ..., x ) - Pn ( x , x , ..., x)l < E sup IPn (t, x, ..., x)l < EM. Hence ~ER IP(x)l = IP, (x, X , ..., x)l 2 ((1 - E ) ~ E) M and 1x1 < T I ( & + ) ~/27cs,-~= T(E);E no longer has the same meaning as in Theorem IV; nonetheless, this theorem is now proved.

since s,-,/t, -, 0 as m -+ +a. Now let P be any finite sum a(,,,...,en, exp 27ci -.. entn)X , let A , be the set of all finite sumsI;-l cktk.If n < m, there is nothingto prove: A, is finite, hence A, is harmonious and for each positive E there ) 0 such that for each trigonometric sum P whose freexists a T I ( & > quencies belong to A,, sup IPI < (1 + E ) sup 1 P(t)l.

+

It1 < T ( e )

R

If n

+

> m , a trigonometric sum Pn is defimd by

6. High frequencies and independence; complements 6.1. The exponent 2 which appears in all our computations arises for the first time in the crucial Lemma 2 which, by iteration, gives Theorem 111. It is a fascinating problem to replace the frequency v by a finite set vl , ...,vn of real numbers which are independent over Q and to ask the following question : Does a constant C exist such that for each sufficiently smallpositive I and each sequence a. , al , ..., an of comk;lex valued bounded continuous functions on the line whose spectra lie in [0, I],

+ .. + an(x)exp 27cixnI + an(x)exp 2nivnxl? < (1 + ~ 1 ~ 'sup " ) lao(x) + al(x) exp 2nivlx + R (8.17)

sup lao(x) + al(x) exp 2nixl and it will be proved that for a suitable T(E),sup (PnI < ( 1 R"

+ E)

Rn+1

1'1

SUP 6T ( e )

IP(t)l, which is an improvement of Theorem IV. Let a, B,, ...,r6, be n - m + 2 real numbers such that

Since A , is finite and since for fixed P,, .. , In, the spectrum of x -+ Pn (x,Pm, .., Pn) lies in A,, n can be replaced by a y, lyl < TI(&), such that IPn (7,P n , Pn) 1 > ( 1 - E ) SUP IPn ( x ,Pm,. ,B n ) I 2 ( 1 - E)' M. 3

R

Consider the interval [y,y

-

+ ~/27cs,-, J; its length exceeds

The answer is very surprising: THEOREM V: With the preceding notation, estimate (8.17) is the best possible in the sense that 12/" cannot be replaced by a number which is o(12/")when 1 tends to 0; moreover this estimate (8.17) holds ifand only i f for some constant D > 0, each integer m > 1 and each sequence p i , ... ,pn of rational integers satisfying I pjl < m , (8.18) lplvl + ... pnvnl 2 Dm-"+ l .

+

Roughly speaking, Theorem V asserts that the best estimate in (8.17) for n indeperldent frequencies holds if and only if these frequencies are strongly independent.

8

250

ULTRA-THIN SYMMETRIC SETS

COROLLARY: Let v, , ...,v, be n real numbers satisfying (8.18) and let (tk)k>lbe a sequence of positive real numbers tending to 0 sufficiently rapidly that (tk+l/tk)2/n< + co. Let En be the set of all sums for E , = O or v l , ..., or v,. Then En is a set of synthesis and a set of uniqueness and all the restriction algebras A@,) are isomorphic. The proof of the corollary is an easy transcription of the proof of Theorem I. Now to the proof of Theorem V.

z,,

I,,

6.2. Proof of (8.18) => (8.17). As in Lemma 1 , we can confine our attention to the case where the functions a,, .. ., a, vanish at infinity; we put

+ al(x)exp 2niv1x + + a,(x) exp 2nivnx f ( x ,y) = a,(x) + a,(x) exp 2nivl y + ... + a,(x) exp 2nivny

ULTRA-THIN SYMMETRIC SETS

25 1

for sufficiently small I and assume that 1x1 2 mT Let T = (the exact value of m 2 1 will be given in a moment); we get la (1x)l < 1 - am212'". Testing (8.17) on f(x) = a (Ix) P(x), where P(x) = co + C: cj exp 2nivix, we get sup [ sup

(P(x)l,( 1 - am212/")((P((,]2 ( 1 - Cl2In)I(P(I,.

1x14 m T

If am2 > C, the only possibility is

sup IP(x)l 2 ( 1 - Cl2In)IIPll m;

Ixl$mT

Theorem IX of Chapter I applies now give (8.18). The same argument shows that 12'" cannot be replaced in Theorem V by any number which is o(l2/")as 1 tends to 0.

f ( x ) = a,(x)

and M = sup I f(x, y)l. R2

Theorem IX of Chapter I shows that for each positive E and each interval I of real numbers of length C E - ( " - ~ )and I ~ for a fixed x,

Hence there exist two real numbers x, and yo such that Ix, - yO1 < c E - ( n - 1 ) / 2 and I f(x,, yo)l 2 ( 1 - E ) M . As in the proof of Lemma 1 , we have

7. From the group R to the group Dm

It is indeed remarkable that in the heart of the group algebra A(R), dwell other group algebras. For the sake of simplicity we shall confine our attention to the product group Dm of a countable number of copies of 2 / 2 2 and it will be proved that, for each ultra-thin symmetric set E, A(Dm)is a closed subalgebra of the quotient algebra A(E). 7.1. The group Dm and the group algebra A(Dm) Let D be the group with two elements 0 and 1 where addition is defined modulo 2 and let Dm be the product group D, of a countable set of

n

k>l

Multiplying f ( x o , yo) by a constant of absolute value 1 if necessary, we may assume that f(x,, yo) is positive. Then, putting A = a + ib, we get

+

hence ( x - x,) a < EM 2n212( X - x , ) ~M . By an optimal choice of x - x,, la1 < 2521 & M ; (8.19), with this estimate of a, gives 1 f(y,, yo)l 2 ( 1 - Cll2In)M when E = I2In, which is (8.17). Proof of (8.17) => (8.18). Let a : R -, C be a function in the class Y of Schwartz whose Fourier-transform is a positive indefinitely differentiable function supported by [0, 11. Assuming that a(0) = 1 , we get, for two positive constants a and j3, 0 < j3 < 1 , la(x)l < 1 - ax2 if 1x1 < 1 and la(x)l < j3 if 1x1 2 1.

copies of D. An element o of Dm is a sequence ( w , ) ~ of ~ elements of D; addition is componentwise addition modulo 2. The topology of Dm is that of componentwise convergence and the Bore1 field of Dm is generated by elementary sets E (a,, ...,a,) of the type ol= a,, .. .,con = a,. The Haar measure d o of Dm is defined by assigning to E ( a , , ..., a,) the measure 2-". For each n 2 1 , let xn : Dm + T be defined by xn(o)= (x, is a continuous character on Dw and each character x : Dm + T can be written x,,(o) = x;' X> for a suitable n 2 1 and a suitable sequence y = (a,, ..., a,) of 0's and 1's. The dual group of Dm is the direct sum of a countable set of copies of D. We denote by A(Dm)the Banach algebra of all complex valued continuous functions f : Dm + C whose Fourier series is absolutely convergent

-

r

252

ULTRA-THIN SYMMETRIC SETS

253

ULTRA-THIN SYMMETRIC SETS

7.2. A(Dm) as a subalgebra of A(E) Throughout this section, E is an ultra-thin symmetric set of real numbers. THEOREM VI : Let 17: E + Dm be the continuous surjective mapping dejinedby17(x) = o ifx = E?&,tk, E, = 0 or 1 and o = wk = 0 or 1, are connectedby wk = ezk-, E,, (mod2) for all k 2 1. Then for any complex valued continuousfunction f : Dm + C the following two assertions are equivalent: (a>f E 4 D m >; (b) f o 17E A(E). Before proving this theorem, some remarks should be made. Let H be the homeomorphism between E and Dmdefined by ~ ( x k I, &ktk)= (ck)kr 1 when ek = 0 or 1, k 2 1. Then H does not induce an isomorphism between A(E) and A(Dm). Using general theorems ([15], p. 78, th. 4.1.3) on homomorphisms of group algebras it can be proved that such an isomorphism between A(E) and A(Dm) cannot exist: A(Dm) is not isomorphic to the whole of A(E) but only to a closed subalgebra of A(E).

g = (ck)kal such that 82,-1 = E 2 k (k 2 1); G is clearly another copy of Dm and the Haar measure dm of G is a Radon measure on D ~ . For each g, E C(E), we define a continuous function F : Dm + C by F o H = g,, where H is the canonical homeomorphism1 between E and Dm, arid we put L(9) = y if, for each o = P(E),

+

7.3. Proof of (a) => (b) It suffices to show that there exists a constant C such that for each character xv: Dm + T, IJxY0 1711,(,, < C. If xY(o)= x:' ,:x g = xYo 17is defined on E by

.

g(x) = exp in (PI&, $.

.-- + /32n~Zn)when

m

x

=

1&,tk 1

and where P 2 k - = PZk= akfor 1 6 k < n. Now Proposition 2 immediately gives the result.

Plainly L (f o 17) = f for each f E C(Dm)since in this case F is constant modulo G . To show that, for each g, E A(E), y E A(Dm), we can confine our attention to the case where g,(x) = elnx if it can be proved that IlyllacDm)< C for some constant C not depending on n. But, in this case, F is defined by F(E) = einekfk and y(o) = yk(ok),where yk(0) ein ( t 2 k e l + t 2 k ) ) and yk(l) = (e1nf2k-1 einfZk ). For each func=+(I+ tion h : D + C, IlhllAcD,= Ih(0) h(1)l 4 Ih(0) - h(l)l. Hence IIY~IIA(D-) = I I Y ~ I I A ( D ) < 1 This last inequality follows Il~Ila(om)< from the observation that for any two real numbers a and b, lcos a cos bl + [sin a sin bl < 1; hence $11 + eia + elb + ei(a+b)l $ 11 - ela - eib + ei(a+b)l< 1, which implies IlykllAcD) < 1 for each k 2 1.

n?

n?

We first of all define a linear operator L : C(E) + C(Dm) such that, for each f E C(Dm), L (f 0 17)= f and for each f in A(E), L( f ) belongs to A(Dm). These two properties give the required implicatiorl (b) (a) and L will be used to show how problems of spectral synthesis can be transferred from E to Dm. Let P: Dm + Dm be the continuous homomorphism defined by dw, ~,,(mod2)for P(E) = coifs = ( E ~ )o~= ~( ~~ ~, ) ~ ~ , a n=e2,-, each k 2 1. The kernel of P is the subgroup G of Dm consisting fo all

+

n?

+

+

+ -

+

8. Spectral synthesis fails in A(Dm)

8.1. THEOREM VII : There exists a real function f in A(Dm) such that the closed ideals I, generated by f", n 2 1, are all distinct. The ideal I, is the closure in A(Dm) of the set of all products f"h, h E A(Dm); hence I,,, c I, for all n 2 1 and we have to show that f" # I,+,. It is sufficient to prcduce a pseudomeasure S,, i.e. a distribution whose Fourier coefficients form a bounded sequence, such that # 0 while for all h in A(Dm), (S,, hf"+ l) = 0.

(S,,f")

7.4. Proof of (b) => (a)

+

n?

+

(8.22)

.

For a better understanding of the proof, we present a candidate S, If

f:R -+ R is a regular function with a finite set F of zeros without multi-

x,,,

plicity, a distribution S1 = 6' of is well defined by (f'(1)l f'@)l)-' 6' (x - A) p, where 6' is the derivative of the unit mass concectrated in 0 and p is a measure carried by F. Hence (S, ,f = IAEr(lf'(A)l)-' > 0 while, for all regular h, (S, , hf 2, = 0. Unfortunately S, is not a pseudomeasure.

+

H i s defined in Section 7.2

>

ULTRA-THIN SYMMETRIC SETS

256

10. Counter-examples. A random set due to Salem 10.1. THEOREM IX: For each p > 2 there exists a sequence (tk),>1 of positive real numbers such that (tk+1/tk)' < i-co and such that the compact set E of all sums 1 E , = 0 or 1 has thefollowingproperties:

xk,

257

ULTRA-THIN SYMMETRIC SETS

Ekal

(a) E is a set of multiplicity; (b) there exists a complex valued bounded continuousfunction y : R -t C whose spectrum lies in E and which cannot be approached, in the topology of uniform convergence on compact subsets of R, by a bounded sequencey , k 2 1 , of trigonometric sums whose frequencies belong to E (here, bounded means: sup sup Iyk(t)l < + a).

negligible set N j . Let N be the union of all the N j ; the measure of N is 0 and for o # N a n d j 2 1, lim jf, < 1 which implies lim f,(o) = 0. k++m

k-+a,

To obtain the estimate (8.26), we apply the following lemma: 5: There is a constant C such that, for each integer m 2 1 and LEMMA each interval I of real numbers whose length III exceeds n/2,

,

k>l

teR

10.2. Let 0 < a < 3 be such that a p > 1. Each t, will be chosen at random and independently of the others in the interval I , = [(k!)-", 2 ( k !)-"I ; then (tk+ Itk)' < + co and for almost all choices of the sequence o of such t,, the corresponding compact symmetric set E ( o ) o all z y e k t k ,E , = 0 or 1 will have the required properties. Let Q be the compact product space I,, dt, the Haar measure of I , normalized by jIkdt, = 1 and d o the product Radon measure on Q: d o = dtl Q ... @ dt, @

xk2

nk,

a * . .

10.3. To prove that E ( o ) is a set of multiplicity, we use the function y ( x , o) = Icos xtkI as we did in Chapter 111, Section 3.3). If ol is sufficiently close to 5, which may always be assumed, E ( o ) is contained in [0,n[ and to prove that y ( x , o)vanishes at infinity, it suffices to consider only integral x. Let F,, k 2 1, be the finite set of integers x such that 2n(k!)" < x < 2n ( ( k I)!)" and letf k ( o ) = sup y ( x , a).To prove that lim f,(o)

n?

+

XPF~

k-

= 0 for almost all o in Q, the following lemma will be needed.

+m

Proof: By pericdicity afid symmetry the problem can be reduced- ~to2 / the 2. in case where I = [0,n/2]. For a suitable I > 0 , we have cos x < e [O, 11 and for any I, cos x < cos 1 in the remainderinterval [I, n/2].Hence cosmx dx < e-mx2'2dx + (42) cosmI Jn/2m,whichgives(8.27). The integral of a product of independent random functions is the product of the integrals. Hence j, ym( x , m) d o = jIklcos x tklmdt, < ,j lcos xtjlmdtj. If x E F,, each of the integrals in the last product is bounded by cm-lI2. Therefore ym ( x , co) d o C Clearly jDf :(a) d o C ym ( x , a) d o C (Card F,) c k m - ' I 2 < 2n ( ( k + I)!)" ~ , m - ~ Choose '~. m = m , = k and then for X > 1 we have Xmkc k m k ( ( k + I)!)" < + co which, by Lemma 4, finishes the proof. To prove the remaining statements of Theorem IX, the additive properties of the random sets E ( o ) must be described.

-

Jr2

nt

ny

l, xxsFk SO

xk,

10.4. Additive properties of the random symmetric sets E ( o )

DEFINITION 4: Let (tk)k,l be a sequence of real numbers such t h a t x y Itk( < + co. We say that (t,),, is fully independent if for any bounded sequence ( P , ) , ~ of rational integers the following implication holds:

LEMMA 4: Assume that an increasing sequence m , of integers tending to injinity exists such that for each X > 1

Then,for almost all o in Q, lim f,(co) = 0. k-r

+ co

x,,

Proof of Lemma 4: Let gj(w) = jmk f ,m" (a);g j ( o ) E L1 (Q, d o ) and so each gj is finite almost everywhere, i.e. for all o not belonging to a

It is easy to show that full independence implies lim -t,+,/t, = 0 but a Q-linearly independent sequence (tk)k21of real numbers such that lim tk+l / t k = 0 is not always fully independent. The terminology of the following proposition is that of Theorem IX: I , = [(k!)-", 2(k!)-&]and o = (tk)krl is a random choice of t , in I,. PROPOSITION 5: For almost all o in Q, (t,),,

is fully independent.

258

ULTRA-THIN SYMMETRIC SETS

ULTRA-THIN SYMMETRIC SETS

Proof: Let N be the set of all o in SZ for which there is a bounded sequence (p,),. of rational integers such that x y pktk = 0 ;if furthermore, such a sequence (pk),,, exists with lpkl < m for all k > 1, we write o E N(m) : Nis the union of all such N(m). If o E N(m),there is a smallest value j of k for whichp, # 0 and we then write o E N (m,j ) : N(m) is the union U N (m,j). If all these N (m, j ) have measure zero, so does N acd

,

1

j> 1

Proposition 5 is proved. Each N ( m , j) is a closed subset of 52 and the measure of N ( m ,j) will be computed by means of Fubini's theorem. Each o in N (m,j ) can be written as a pair ( o r ,a"),where of= ( t , , ..., tj) and o " = (tj+,, ..., t,, ...). If for each fixed o" in Q" = j+ I, the measure, computed in = Q f , of all of such that (of, o") E N (m,j)) is 0, so is the measure of N (m,j). But the statement ( a ' , a") E N (m,j ) is one of the 2m statements

nl,k,j~k

nk,

I

/

xk,

where G(o")is the set of all sums j+ pktk for all integral pk such that IpkI < m. Since tk+1/tktends to 0. the measure of the symmetric set G(otf)of real numbers is 0 and so is the measure of the set of tj satisfying (8.28).

El,@

< CIISlla*

and S,

-+

S in o (W*, 9). (8.30)

We can state PROPOSITION 6: For almost all o in Q, A (E(o))is of type 2. This at once finishes the proof of Theorem IX: in the situation of restriction algebras A(E) to compact sets E of real numbers, (8.30)is equivalent to $,(x) -+ $(x) uniformly on compact subsets of R (8.31) and

Then for almost all o in Q, the following equality holds for each complex measure p carried by a finite subset of E ( o )

xl,@

Before we conclude the proof, we shall describe the difference between the Banach algebras A(E) and A(E,). It is unknown whether a restriction algebra to a set of multiplicity can be isomorphic to a restriction algebra to a set of uniqueness. Such a theorem would prove that for almost all a E Q, A ( E ( o ) )and A(E,) are not isomorphic. The method we shall use is the following: let 9 be a regular semisimple commutative Banach algebra which will be identified, by means of Gelfand's representation, with a function algebra on the spectrum K of 9 ;the pseudo-measures are the elements of the dual space W* of W and the support of such a pseudomeasure can be defined in a natural way. Let o (W*, 92) be the weak-star topology. We say that 92 is of type 1 if each pseudo-measure S is the weak-star limit of a sequence S,,, n >, 1, of pseudomeasures carried by finite subsets of K. If this is not the case, we say that 9 is of type 2. General theorems of Banach give the following information on Banach algebras of type 1 :there is a constant C > 0 such that for each S in W*, a sequence S, of finitely supported pseudomeasures can be found such that

IISnll,*

,,

COROLLARY : For B > 3, let ( s ~ ) be ~ a fully independent sequence of real numbers chosen in ~ ( k ! ) -2(k!)-'1 ~, and let El be the ultra-thin symmetric set of all sums xk,l E ~ S L E, , = 0 or 1. Let H be the homeomorphism from E ( o ) to El defined by

Proof: Let 8 be the additive group generated by E : @ is the set of all sumsxk>,pktk corresponding to all bounded sequences (p,),. of rational integers; let 8, be the additive group generated by E l . The full independence of the sequences (t,),, and ( s ~ implies ) ~ ~that Hcan be extended by H (Xkalpktk)= Yk.,pksk to an isomorphism, still denoted by H, from 8 to (3,. If P is any finite trigonometric sum, P(t) = c(1) exp 2nih and if G(t) = c(A) exp 2niH(iZ)t, Kronecker's theorem (Chapter I, Section 5) gives 11 Pll oL, = 11 Q 11 m, which is (8.29).

10.5. Last part of the proof of Theorem IX

as we have already observed in Chapter VII, Section 3.2. The following lemma will be useful in proving Proposition 6: i

1

LEMMA 6: Let f , : R -+ [O, 11 be defined by f,(x)

=

cos t l x ..-cos tk-,x sin tkx cos t,+,x

Then for almost all o in 52, sup I fkl -, 0. R

cos t,x

..-

260

ULTRA-THIN SYMMETRIC SETS

ULTRA-THIN SYMMETRIC SETS

The proof is an easy modification of that given in Section 10.3: the term sin t k xinstead of cos t,x ensures uniform convergence to 0 on compact subsets of R. Note that f , = for a suitable measure p, carried by E.

THEOREM X: For almost all o in 9 , A ( o ) is not a coherent set of frequencies. + t,) x c o s n t , ~ c o s n t , ~ ; Proof: Let P, ( x , o) = exp n i ( t , + the spectrum of P, lies in A and Theorem X will be proved if, whenever 0 < a < b there exists a negligible subset N (a, b) of SZ such that, P, ( x , o) converges uniformly to 0 on [a, b] for o 4 N (a, b); then, if o does not belong to the negligible union of all the N ( j - l , j), j 2 1, A ( o ) is not a coherent set of frequencies. Let F, be the finlte set of all x = j (kt,)-l, j E N , lying in [a, b]. For each x in [a,b ] ,there is a y in F, such that Ix - yl < (2ktk)-I and Bernstein's inequality (Chapter V, Section 4) shows that IP,(x) - P,(y)( < C k - l , where C does not depend on k . Hence it suffices to prove that f k ( o ) = sup IPk (x, o)I tends to 0 for almost all o in 9.The remainder of

LEMMA 7: The pseudo measure norm of H(p,) does not tend to 0. Proof: By Theorem 11 this norm is greater than C-l sup cos x , . . cos x k - sin x , cos xk+ ...,where C is a positive constant and ( x , ) ~ , runs over all sequences of real numbers. But this last sup is 1. Proof of Proposition 6 : Assume that A (E(o)),which we shall write A(E), is of type 1 and that o does not belong to the union of the two negligible sets which arise in Lemma 6 and Proposition 5 respectively. Let (,u,,~)~,,be a sequence of measures carried by finite subsets of E and satisfying IIpk.jllpM < CllpkllpMand (pk,j , f ) -, ( p k , f )for all locally constant functions f on E; such a sequence exists by (8.30). Then by Corollary of Proposition5 ( [ H ( , ~ ~ , = j ) ( (l p~k~, j ( ( p M acd clearly (H(P,,~), f ) -, (H(pk),f ) for all locally constant functions f on E l . Hence H(p,) is the weak-star limit of the bounded sequence of pseudomeasures H ( p k S jj) , 2 1, since the locally constant functions are dense in A(El), and since norms decrease in the passage to weak-star limits l l H ( p k ) l l< ~ ~lim IIH(pk,j)ll < Cllpkllp~.Lemmas 6 and 7 show that

,

j++m

this is not the case.

11. Other random sets Let (tk)k,l be an increasing sequence of positive real numbers such that t,,, 2 3tk (for all k 2 1 ) and let A be the set ofall finite s ~ m s z ektk, ~ , ~ E , = 0 or 1. Theorem IV asserts that if (tk/tk+l)2< + co,A is a coherent set of frequencies. We shall prove that the exponent 2 cannot be replaced by a greater one and that a sequence (t,),,, can be chosenin such a way that (tk/tk+ < while A is not harmonious. In both cases random methods will be used. Let 0 < ol < & let I,, k 2 1 , be the interval (k!)" < x < 2(k!)", let dt, be the Lebesgue measure of I, normalized by j,, dt, = 1, let SZ be the compact product space I, and let d o be the product Radon measure on SZ defincd by d o = 8 dt,.

I,,,

xkg

+

nk,

k b1

For each o = (t,),, in SZ, let A(@)be the corresponding discrete set of sktk,E , = 0 or 1. all finite sums

xk,

26 1

xeFk

the proof is an easy transcription of that given in Section 10.3. If 4 < ol < 1 the symmetric set A ( o ) constructed with (tk)kal,where tk E I, = [(k!)",2(k!)"]is a coherent set of frequencie?. Moreover for each positive E , there exists a T (depending on E and co) such that for each trigonometric sum P whose frequencies belong to A , sup /PI < ( 1 E )

+

R

sup IP(x)(.This property is shared by all harmonious sets. However 1x1

=

THEOREM XI : For 3 < ol < 1 , let I, be the interval [(k!)",2(k!)"]and let o = (t,),,, where each t , is chosen at random and independently of the other tj in the interval I,. Let A ( o ) be the discrete set of allJinite sums cktk,E , = 0 or 1. Then for almost all choices of the sequence of t k , k 2 1 , A ( o ) is not harmonious. (To be more precise we recall that each t, is equidistributed in I,.)

xk>l

This theorem is very sharp since, following the method given in Chapter 11, Section 4, it can be shown that i f T k , , tk/tk+ < co A is harmonious. We shall prove that, if 0 < a < b there is a negligible subset N1(a,b) If o does not of SZ such that, for o 4 N (a, b), inf sup lllxll 2

+

A.

a<x
AEA

belong to the negligible union of all the N' ( i - l , j), j 2 1 , A is not harmonious. We recall that for each real number x , llxll is the distance from x to the nearest integer. Assume that Illxll < $j for some x in [a, b] and all A in A. Then &,I JJtkxJJ < (see Chapter 11, Section 8, Proposition 7). Let Fk be defined as in the proof of Theorem X, let Sk (x,o ) be lltlxll lltkxll

+ +

ULTRA-THIN SYMMETRIC SETS

262

and Pk(x,o ) = (1 - Iltjxll). For each x in [a, b], there is a y in Fk such that Ix - y] < (2ktk)-l; hence Sk(x, o) < & implies Sk(y, o ) < ) for k 2 ko. The elementary inequality (I - aj) > 1 - aj, (0 < aj < 1) gives Pk(y, o ) > $ for k 2 k, and y E Fk. But an easy modification of the argument used in Section 10.3 shows that sup Pk(JI, a ) Fk + 0 for o $ N' (a, b). We have thus got a contradiction.

n:

C:

12. Notes The main idea of modern harmonic analysis is that, when we restrict our attention to 'ultra-thin sets', all becomes clear and obvious, all Banach algebras are isomorphic.. . This feeling was tested on Kronecker sets (A(E) = C(E) trivially in this case) and on Helson sets (A(E) = C(E)). To get more interesting restriction algebras, it is necessary to consider ultra-thin sets with a given additive structure: if K, and K, are two compact subsets of the line, K, n K2 = 0 and Kl u K2 is a Kronecker set, then A (Kl + Kz) = C(K,) Q C(K2) the tensor product being a projective one as Varopoulos showed [93]. Ultra-thin symmetric sets were studied in [68] and [86].

-

-

CONCLUSION

Do connections actually exist in harmonic analysis between additive properties of a set A of frequencies and functional properties of corresponding functions constructed with these frequencies? No general answer can be given. Throughout this book, both positive and negative results are to be found. Positive results Relatively dense harmonious subsets of a locally compact abelian group were characterized by three equivalent properties: the first concerns Diophantine approximation, the second concerns additive properties, and the last one deals with functional properties. (The reader is referred to the Interlude.) Rapid prediction of the size of trigonometric sums whose frequencies belong to a finite set F of real numbers is equivalent to the strong independence of F over Z (Theorem IX of Chapter I). Negative results In Chapter VIII, ultra-thin symmetric sets E and a random set F defined by Salem were studied. It may happen that E and F have the same additive structure (a homeomorphism H: E F exists which can be extended to an isomorphism between the additive groups generated by E and F). However, E and F do not give rise to similar functional properties: bounded continuous functions whose spectra lie in E can be nicely approximated by trigonometric sums whose frequencies belong to E; this is not the case for F. Open problems It is not known whether coherent sets of frequencies or Sidon sets can be characterized by additive properties.

SOME O P E N PROBLEMS ABOUT S Y M M E T R I C SETS

O P E N PROBLEMS

Special trigonometric series on IocalJields E is a symmetric set constructed with

4 constant dissection ratio

0 Salem

=

0-I

Other 8

Variable dissection ratios E,, k 2 l Random set due to Salem ( E k ~ k - ~

nd 0 < a < +

E is a set of uniqueness

no

E is a set of synthesis

?

A(E) is of type 1 Bochner's property holds

no

no

Let K be a local field1 of characteristic 0. A standard subset A of Kwas defined (Chapter 11, Section 13) in the following way. Let k be an algebraic number field and n : k + K an isomorphic embedding with a dense range. Consider the set V of all possible metrics on k (metrics are either defined by valuations or real or complex absolute values); for a v E V, ',xlvwill denote the value of the corresponding metric at x. Let v, be the metric induced by that of K. Then A is the set of all n(x) for all x in k such that lxl,, < 1 for all v # v, in V. Three examples of standard sets have been thoroughly studied in this book or in classical harmonic analysis: (a) A = Z and K = R (then k = Q); (b) k is a real quadratic field and A is the set of all quadratic integers x in k whose conjugate lies in [- 1, I]; note thatA also contains 1 and - 1. (c) K = Q, and A is the set of all mp" in [- 1, 11 (m E Z. n E Z). The last two cases are very similar: let E(A) be the space of all almost periodic functions on K(identified with the dual group) whose frequencies belong to A. There exists a critical positive m such that for any Riemann integrable compact subset C of K (1) if the Haar measure of C exceeds m , sup If 1 and Sup If 1 are equiC K valent norms on E(A); (2) if the Haar measure of C is less than m, this is not the case. It is to be expected that this result holds for all standard sets in local jields of characteristic 0 with the exception of the discrete subgroups of R and C.

It is assumed that K is not discrete.

265

267

APPENDIX I

APPENDIX

has been reached. Hence an integer m and a sequence B = B,, B, , . .., Bm of finite subsets of G exist such that IA + Bj+,l - IA + Bjl = IBj+ - lBjl > 0, (0 < j < m - 1 ) and such that the complement of A + Bm is (0). Adding all these equations, we get IA + BmI - IA + BI = lBml - IBI or IA BI = IA Bml - lBml IBI. Bm satisfies the conditions of the first case. Hence IA + BmI > IAI lBml - 1 and so IA BI > IAI + IBI - 1. Now Lemma 1 has to be proved. Let n 1 , n > 1, be the Cardinal of C ; C = (0, c,, . .., c,). A first observation is: one of the sums c, c j , 1 6 i < n, 1 < j < n, belongs to A B. If this were not the case, C would be closed under addition. Since p is a prime, there are only two possibilities: C = {0), which was our first case, and C = G, in which case A and Bare empty and Theorem I is trivial. We thus have a relation (4) c + d = a + b , a ~ A , b ~ B , c $ A + B , d $ A + B . From now on a relation such as (4) is chosen and b is fixed. Let B' be the set of all b - c j , 1 6 j 6 n, for which an element a' of A exists such that a r + b - c j belongs to C : a ' + b - c j = c k , 1 < k < n . Let B1 = B U B'. We list some properties of B': (5) B' contains b - c, where c is given by (4);hence B' is not empty; ( 6 ) B' and B are disjoint: if b - cj = b', then c, = a' + b' E A + B, which is not the case; (7) 0 4 A B'. If a" b - cj = 0, cj = a" b E A B whichis not the case; (8) ( A B') n C = b - B'. If c, E A B', then c, = a" b - cj, which can also be written cj = a" b - c,; by our definition of B', b - c, belongs to B' and c, belongs to b - B'. Hence ( A + B') n C c b - B'. On the other hand, b - B' c ( A + B') n C. In fact, let x = b - cj be in B'; then we can find an a' in A and a c, in C such that a' + b - cj = c,; cj = b - x belongs to C and is a sum a' b - c,. Hence cj also belongs to A + B'. (9) The complement of A B in A B1 is b - B'; we have to show that ( A B,) n C = b - B'. But A B, = ( A + B) v ( A + B') and ( A + B) n C is empty while ( A + B') n C = b - B'. (10) Hence IA B,I - IA+ B( = Ib - B'I = lBll - 1B1. This last equality holds because B and B' are disjoint.

+

+

+

A. Additive problems

I

From now on, for any subsets A and B of a group G, A + B will denote the set of all sums x y, x E A, y E B and IAI will denote the Cardinal of A. THEOREM I :Let p > 2 be a prime and let G be thefinite group Z / p Z with p elements. Let A and B be two subsets of G. Then either A B = G, o r ( A BI > IAI IBI - 1. Proof: It will be shown that A B # Gimplies IA BI > IAl IBI - 1 and the idea of the proof is to enlarge B in such a way that the complement of A B is reduced to a single element of G. It should be noted that if B is replaced by c + B, c E G, A B is replaced by c A B. This substitution does not affect the required B Z G, it may be assumed that 0 4 A + B. conclusion. Hence if A Two cases may occur: Case 1 : The complement of A B in G is equal to 0. Then IA BI = p - 1. On the other hand - A and B are disjoint subsets of G. Hence ]-A1 1B1 < p, which implies IA + BI > IAl IBI - 1. Case 2: The complement C of A B in G is not equal to 0.

+

+

+

+

+

+

+ +

+

+

+

+

I

+

+

+

+

+

+

+

+

+

+

LEMMA 1: In the second case, a subset B, of G can be found with the following three properties: ( I ) B c B, c G; (2) IA BlI - IA + BI = lBll - 1B1 > 0; ( 3 ) 0 4 A + B, . Before proving this lemma, it should be explained why it implies Theorem I. Using this lemma inductively, a strictly increasing sequence B, c B, . . . of finite subsets of G can be constructed. Since G is finite, the process will stop. But the process only stops when the situation of case 1

+

+

+

1

+

+

+

+

+

+

+

+

+

I

+ +

+

APPENDIX

APPENDIX

268

THEOREM I1 :For n 2 1 , let T nbe the n-dimensional torus and let a be the Haar measure of T n normalized by o(Tn) = 1. Let A and B be two compact subsets of Tn. Then either A B = T n or a ( A + B) 2 a(A) + o(B). The idea of the proof is to reduce the problem to finite ones. be a sequence of finite subgroups of Tn such that Let (a) the number of elements in rkis k ; (b) if o , is the Haar measure of rknormalized by o k ( r k )= 1 , o is the weak-star limit of these o,, k 2 1 . We assume initially that A and B are Riemann-integrable. Then ak(A)-' o(A), ak(B)+ o(B) as k + + CO. We have ( A + B) n rk= A n rk+ B n rkand for all primes p, we have either (11) A n r , + B n r , =r,,or (12) o p ( A n r p B n r , ) 2 o p ( A n T p )+ o P ( B n r p )- l / p . If ( 1 1 ) holds for an infinite number of primes p, A + B = Tn. On the other hand, if (12) holds for an infinite number of primes p, o ( A + B) 2 o(A) o(B). For arbitrary compact subsets A and B, we construct a decreasing sequence A, (resp. B,) of Riemann-integrable compact subsets of T n such that for each x E A, (resp. x E Bk), d ( x , A) < l / k (resp. d (x, B) < llk). If A + B is not the whole of Tn, A, + B, is not the whole of Tn for k 2 k o , say. Hence a ( A , + B,) 2 a(Ak) + o(Bk)for k 2 ko and passing to the limit, a ( A + B) 2 a(A) + o(B) which concludes the proof of Theorem 11.

+ (rk),,,

+

+

B. Kronecker's theorem THEOREM 111:Let a,, ...,a, be n real numbers linearly independent over Q. For each sequence xl , ..., x, of n real numbers and each positive E, there exist a real number t and n rational integers p, , ...,p, such that ( x , - colt ~ ~ < E1, 1 Ixn - ~ n -pnI t <E Proof: Let r b e the subgroup of all x = (x,, . . . ,x,) in Rn such that we can write x j = wjt + pj for some tin R and rational integerspj, 1 < j
2: Let H be a closedsubgroup of R". Then either H = Rn or there LEMMA exists a linear form I on Rn such that 1 # 0 and I(x) E Z for all x E H. Proof of Lemma 2 : If H i s not the whole of Rn, an integer m 2 0 , m < n and a basis e l , .. ., en of Rn can be found such that the equations of H

I

269

, E Z , . . ., X , E Z. = x , = 0, ,,x with respect to this basis are x , = It is now sufficient to put l(x) = x,. Returning to Theorem 11, if the closure of 7 is not the whole of R", a linear form I # 0 exists such that l(x) E Z for each x in T . Testing on x E Zn, we find that l(x) = qlxl + ..- + q,xn for rational integers q,, ..., qn and testing then on x = (colt, ..., w,t), we find colql + ... + w,qn = 0, which contradicts the hypothesis.

BIBLIOGRAPHY

271

B. Original papers

BIBLIOGRAPHY

A. Basic reference books [I] Banach, S., Thiorie des opirations liniaires. Monografje Matematyczne. Warszawa 1932. [2] Bary, N.K., A treatise on trigonometric series, vols. I and I1 (translated from the Russian). New York, Pergamon Press, 1964. [3] Borevich, Z. I. and Shafarevich, I. R., Number theory (translated from the Russian) New York, Academic Press, 1966. [4] Cassels, J. W. S., An introduction to Diophantine approximations. Cambridge, University Press, 1957. [5] Cassels, J.W.S., An introduction to the geometry of numbers. Berlin, SpringerVerlag, 1959. [6] Edwards, R.E., Fourier series, a modern introduction. New York, Holt, Rinehart and Winston, 1967. [7] Hewitt, E. and Ross, K.A., Abstract harmonic analysis, vols. I and 11. Berlin, Springer-Verlag, 1963 and 1970. [8] Kahane, J.-P., Siries de Fourier absolument convergentes. Berlin, Springer-Verlag, 1970. [9] Kahane, J.-P., Some random series of functions. Boston, Heath and Co, 1968. 1101 Kahane, J.-P. and Salem, R., Ensemblesparfaits et s t i e s trigonomitriques. Paris, Hermann, 1963. [ll] Katznelson, Y., An introduction to harmonic analysis. New York, Wiley and Sons, 1968. [I21 Lang, S., Introduction to Diophantine approximations. New York, AddisonWesley Publ. Cy., 1966. 1131 Mann, H.B., Addition theorems. Interscience tracts in Pure and Applied Mathematics, number 18. 1965. [14] Reiter, H., Classical harmonic analysis and locally compact groups. Oxford, Clarendon Press, 1968. [15] Rudin, W., Fourier analysis on groups. Interscience tracts in Pure and Applied Mathematics, number 12. 1962. 1161 Saks, S. and Zygmund, A., Analytic functions. Warszawa and Wroclaw 1952. [17] Salem, R., Algebraic numbers and Fourier analysis. Boston, Heath, 1963. [18] Salem, R., CEuvres mathimatiques. Paris, Hermann, 1967. [19] Samuel, P., Thiorie algibriques des nombres. Paris, Hermann, 1967. [20] Serre, J.-P., Cours d'arithmitique. Paris, Presses Universitaires de France, 1970. [21] Schwartz, L., Thiorie des distributions, vols. I and 11. Paris, Hermann, 1966. [22] Weil, A., Basic number theory. Berlin, Springer-Verlag, 1967. [23] Wiener, N., The Fourier integral and certain of its applications. Cambridge, University Press, 1933. 1241 Zygmund, A., Trigonometric series, 2nd ed., vols. I and 11. Cambridge, University Press, 1968.

I

I

I

I

I

! 1

[25] Arnara, M., Ensembles fermks de nombres algkbriques, Sim. Dubreil-Pisot, Inst. H.Poincare 19, 1 (1965-66). [26] Baker, A., Rational approximations to t/?and other algebraicnumbers. ~ u a r tJ.. Math. 15 (1964), 375-383. 1271 Benzaghou, B., Algebres de Hadamard. Bull. Soc. Math. France, 98, 3 (1970), 209-252. [28] Bertrandias, F., Caractkrisation des ensembles S, par la rkpartition modulo 1 en p-adique. Sim. Dubreil-Pisot 17 (1963-64), nr. 11. [29] Bertrandias, F., Ensembles remarquables d'adeles algkbriques. Bull. Soc. Math. France, Mkm. 4 (1965). [30] Beurling, A. and Malliavin, P., On the closure of characters and zeros of entire functions. Acta Math. 118 (1967), 79-93. [31] Bonami, A., Private communication. [32] Bonami, A., Etude des coefficients de Fourier des fonctions de Lp(G). Ann. Inst. Fourier (Grenoble) 2 (1970). [33] Boyd, D. W., Transcendental numbers with badly distributed powers. Proc. Amer. Math. Soc. 23 (1969), 424-427. [34] Cassels, J.W.S., On a result of Marshall Hall. Mathematika 3 (1956), 109-110. [35] Chabauty, C., Sur larkpartition modulo 1 de certaines suitesp-adiques. C . R. Acad. Sc. Paris 231 (1950), 465-466. [36] Decomps-Guilloux, A., Elkments algebriques dans les adeles. Sim. Dubreil-Pisot 4, 1 (1970). [37] Deschamps-Gondym, M., Compacts associks a un ensemble de Sidon. C . R. Acad. Sc. Paris 271 (1970), 590-592. [38] Deschamps-Gondym, M., Sur les ensembles de Sidon topologiques. C . R. Acad. Sc. Paris 271 (1970), 1247-1249. [39] Domar, Y., Sur la synthBse harmonique des courbes de RZ. C . R. Acad. Sc. Paris 270 (1970), 875-878. [40] Drury, S.W., Sur les ensembles de Sidon. C . R. Acad. Sc. Paris 271 A (20 juillet 1970). [41] Ehrenpreis, L., Mean periodic functions. Amer. J. Math. 78 (1955), 292-328. 1421 Ferru, H., Caractkrisation des nombres de Pisot complexes. Bull. Soc. Math. France, Mem. 19 (1969), 5-20. 1431 Ferru, H., Extension au cas de R ~ . C. R. Acad. Sc. Paris. (This paper contains the correction of an error in the previous one.) [44] Hartman, S., Rolewicz, S. and Ryll-Nardzewski, C., Ultra-Kroneckerian sets. Colloq. Math. 16 (1967), 225-229. 1451 Herz, C., Spectral synthesis for the Cantor set. Proc. Nat. Acad. Sc. U.S.A. 42 (1956), 42-43. [46] Herz, C., Spectral synthesis for the circle. Ann. of Math. (2) 68 (1958), 709-712. [47] Kahane, J.-P., Lectures on mean-periodic functions. Tata Institute of Fundarnental Research, Bombay (1959). [48] Kahane, J.-P., Pseudopkriodicitk et series de Fourier lacunaires. Ann. Scient. E.N.S. 3rd series 79 (1962), 93-150. [49] Kahane, J.-P., Sur les fonctions moyennes-phiodiques bornees. Ann. Inst. Fourier (Grenoble) 7 (1957), 292-315. [SO] Kahane, J.-P. and Katznelson, Y., Contribution a deux problemes concernant les fonctions de la classe A. Israel. J. Math. 1, 2 (1963), 109-131. [51] Katznelson, Y. and McGehee, O.C., Measures and pseudo-measures on compact subsets of the line. Math. Scand. 23 (1968), 57-68.

..

272

BIBLIOGRAPHY

BIBLIOGRAPHY

[52] Kessler, I., Semi idempotent measures on abelian groups. Bull. Amer. Math. Soc. 73 (1967), 25-8260. [53] ItB, T. and Amemiya, I., Asimple proof of the theorem of P. J. Cohen. Bull. Amer. Math. Sac. 70 (1964), 774-776. [54] Landau, H.J., Necessary density conditions for sampling and interpolation of a certain entire function. Acta Math. 117 (1967), 37-52. [55] Lohoue, N., Nombres de Pisot et synthkse harmonique dans les algkbres AJR). C . R. Acad. Sc. Paris 270 (1970), 1676-1678. [56] Lohoue, N., La synthbse des convoluteurs sur un groupe abtlien localement compact. C . R. Acad. Sc. Paris 272 (1971). [57] Lohoue, N., Thesis, Facult6 des Sciences d'Orsay, 1972. [58] McGehee, O.C., Sets of uniqueness and sets of multiplicity. Israel. J. Math. 4 (1966), 83-96. [59] McGehee, O.C., Certain isomorphisms between quotients of a group algebra. Pacific J . Math. 21 (1967), 133-152. [60] Malgrange, B., Existence et approximation des solutions des equations aux derivees partielles et des equations de convolution. Ann. Znst. Fourier (Grenoble) 6 (1955), 271-355. [61] Malliavin, P., Sur l'impossibilit6 de la synthbse spectrale sur la droite. C . R. Acad. Sc. Paris 248 (1959), 2155-2157. [62] Malliavin, P., Impossibilite de la synthkse spectrale sur les groupes abeliens non compacts. Publ. math. Z.H.E.S. Paris 2 (1959), 61-68. [63] Malliavin, P., Ensembles de resolution spectrale. In: Proc. Znt. Congress Math. 1962. Djursholm, Sweden, Institut Mittag Leffler, 1963, 368-378. [64] Malliavin, P. et Malliavin-Brameret, M.P., Caracterisation arithmdtique d'une classe d'ensembles de Helson. C . R. Acad. Sc. Paris 264 A (1967), 192-193. Malliavin, P. and Beurling, A. (See Beurling, A. and Malliavin, P.) [65] Mela, J.-F., Sur les ensembles d'interpolation de C.Ryll-Nardzewski et de S.Hartman. Studia Math. 29 (1968), 167-193. [66] Mela, J.-F., Sur certains ensembles exceptionnels en analyse de Fourier. Ann. Znst. Fourier (Grenoble) 18 (1969), 33-70. [67] Mela, J.-F., Approximation Diophantienne et ensembles lacunaires. Bull. Soc. Math. France, Mem. 19 (1969), 26-54. [68] Meyer, Y., Isomorphismes entre certaines algkbres de restrictions. Ann. Znst. Fourier (Grenoble) 18, 2 (1968), 73-86. 1691 Meyer, Y., Algkbres de restrictions non isomorphes. Ann. Znst. Fourier (Grenoble) 19 (1969), 117-124. [70] ~ e y e rY:, , Nombres algebriques et analyse harmonique. 4th series, 3, 1 (1970), 75-110. [71] Meyer, Y., Nombres de Pisot, nombres de Salem et analyse harmonique. Lecture notes in math. 117. Berlin, Springer-Verlag, 1970. [72] Meyer, Y., Les nombres de Pisot et la synthbse harmonique. Ann. Sc. E. N . S., 4th series, 3, 2 (1970), 235-246. [73] Meyer,Y. et Schreiber, J.-P., Quelques fonctions moyennes-periodiques non bornees. Ann. Znst. Fourier (Grenoble) 19 (1969), 231-236. [74] Pisot, C., La repartition modulo 1 et les nombres algebriques. Ann. Scuola Norm. Sup. Pisa 7,2 (1938), 205-248. [75] Pisot, C., Familles compactes de fractions rationnelles et ensembles fermes de nombres algebriques. Ann. Sci. E.N.S. Paris, 81, 3 (1964), 165-188. [76] Pisot, C., Quelques aspects de la thtorie des entiers algebriques. Montreal, Quebec, Les Presses de 1'UniversitC de MontrBal, 1966.

[77] Rauzy, G., Algebricite des fonctions meromorphes prenant certaines valeurs algebriques. Bull. Soc. Math. France 96 (1968), 197-208. [78] Rauzy, G., Transformations rationnelles pour lesquelles I'ensemble des nombres de Pisot-Vijayaraghavan est stable. C. R. Acad. Sc. Paris 268 (1969), 305-307. [79] Rauzy, G., Caract6risation des ensembles normaux. Bull. Soc. Math. France 98 (1970), 401-414. [80] Rosenthal, H.P., A characterization of restrictions of Fourier-Stieltjes transforms. Pacific J . Math. 28 (1969), 663-668. [81] Rosenthal, H. P., Projections onto translation-invariant subspaces of LP(G). Memoirs Amer. Math. Soe. 63 (1966). [82] Roth, K.F., Rational approximations to algebraic numbers. Mathernatika 2 (1955), 1-20. [83] Schmidt, W. M., Badly approximable systems of linear forms. J. Number Theory 1 (1969), 139-154. [84] Schmidt, W. M., Simultaneous approximation to algebraic numbers by rationals. Acta Math. 125 (1970), 189-202. [85] Schmidt, W. M., Lectures on Diophantine approximations. Dept. Math., Univ. of Colorado, Boulder, Colo., 1970. [86] Schneider, R., Some theorems in Fourier analysis on symmetric sets. Pacific J. Math. 31 (1969), 175-196. [87] Schreiber, J.-P., Une caracterisation des nombres de Pisot-Salem des corps padiques. Bull. Soc. Math. France, Mem. 19 (1969), 55-43. [88] Schreiber, J.-P., Sur la notion de modkles dans les groupes abeliens localement compacts. C . R. Acad. Sc. Paris 272 (1971), 30-32. [89] Schreiber, J.-P., Thesis, Facult6 des Sciences d'Orsay, France, 1972. [go] Serre, J.-P., Travaux de Baker. Sim. Bourbaki 22 (1969), nr 368. [91] Varopoulos, N., Sur les ensembles parfaits et les series trigonometriques. C . R. Acad. Sc. Paris 260 (1965), 3831-3834. 1921 Varopoulos, N., Sets of multiplicity in locally compact abelian groups. Ann. Znst. Fourier (Grenoble) 16, 2 (1966). [93] Varopoulos, N., Tensor algebras and harmonic analysis. Acta Math. 119 (1967), 51-112. [94] Varopoulos, N., Groups of continuous functions in harmonic analysis. Acta Math. 125 (1970), 109-154.

273

INDEX

almost-periodic functions, 105 almost-periods (or e-almost-periods), 105 badly approximable real numbers, 11 coherent set of frequencies, 104 dual group, 41 Fourier transform (of a distribution), 82, 99 Fourier transform (of a function), 98 frequencies, 105 fully independent sequence, 256 harmonious set, 37 minimal polynomial, 25 model, 48 multiplicity (set of), 81

p-adic field, p-adic valuation, 60 Pisot numbers, 25, 65 produced by (H produced by G), 49 pseudo-function, 82 pseudo-measure (norm), 99 relatively dense, 44 Salem numbers, 34, 65 spectrum, 87, 99 strongly independent, 16 spectral synthesis, 213 synthesis (set of), 216 test-function, 99 trigonometric sum, 105 uniqueness (set of), 82