A basis in the space of solutions of a convolution equation

A BASIS IN THE SPACE OF SOLUTIONS OF A CONVOLUTION EQUATION V. V. Napalkov

i. Introduction. Let D be an arbitrary convex region in the complex plane C; H(D) be the space of analytic functions in D with the topology of compact convergence; G n be an in0 0 creasing sequence of convex compacts in D exhausting D, and G n c Gn+ I (n = i, 2 . . . . ) (G n is the set of interior points of the compact Gn). We introduce functions hn(8) = k n ( ~ ) , 0 < O ~ 2~ (n = i, 2 . . . . ), where kn(O) = sup Re (z exp iO) is the support function of the compact G n. It is known (see, e.g., [i, z~

n

pp. 77 and 96]) that the Laplace transform establishes a topological isomorphism between the strong dual H*(D) [of the space H(D)] and the space of the entire functions P = lim m ind Pm, where

Let F ~ equation

H*(D).

A functional F determines

in the space H(D) the homogeneous

convolution

31v [y] (z) - - (F, Y (z + t)) = O. If r

is the characteristic

(1)

function of Eq. (i), i.e~

r (~) = L (F) (X) - - (F, exp (Xz)). then by (11, m I) . . . . . (lj, mj) . . . . we denote the sequence A = {lj}j=1 ~ of zeros of the function #(I) with respective multiplicities mj. By W we denote the set of all solutions of Eq. (i) in the space H(D). Then the system

l i ! j ~mt-I l zVexp@iz)

E= b e l o n g s t o W.

L e t G be t h e c o n j u g a t e d i a g r a m [2] o f t h e f u n c t i o n

r

and D be t h e maximal

subregion of D that is covered by all possible shifts G~ of the compact G through vectors =, provided G~ c D. The following result is well known in the theory of convolution equations [3, p. 357]. if zeros {lj} of the functiQn r constitute a regular set,% then the system {exp(ljz)} is a basis in the space of solutions of Eq. (i) analytic in D; it can be shown that this system is also a Schauder basis. And if the function ~(I) has a completely regular growth, then

m]--1

~- W we can uniquely assign a series ~j=1 ~v=0 to every function f(z)-=

.ej~zPexp (~jz) whose par-

tial sums (arranged in a certain way) converge to f(z) in the region D; the system {zP exp (Xjz)} will already not be a basis in the general case. This system serves the purpose of proving that, with the proviso that ~(I) is a function of completely regular growth, in the corresponding space W there always exists a Schauder basis composed of linear combinations of elements of the system E. To construct this basis, the results of [4] are used; in doing *A set {bk} is called regular if (I) the set {bk} has a finite density (limr_~(n(r)/r) < =, n(r) is the number of points {bk} in the disc Ill < r); (2) for some constant c there exists a finite limit: '

(3) for some ~ > 0 the inequality

-

- - I b ~ . J < r

'

'

Ibk+ll - tbkl > ~.

Bashkir Branch of the Academy of Sciences of the USSR. Translated from Matematicheskie Zametki, Vol. 43, No. i, pp. 44-55, January, 1988. Original article submitted March 3, 1987.

-0001-4346/88/4312-0027

$12.50

9

1988 Plenum Publishing

Corporation

27

so, it turns out that the condition of "slow decrease" of the function r (which has been used in [4]) is, within the framework of the present article, naturally related to the con-' dition of completely regular growth of r 2. Preliminary Results. In what follows, it will be assumed that a function r with an indicatrix of growth h(O) has a completely regular growth and a region D coincides with the union U=~QG=, where Q is some convex region and G= is the shift by a vector = of the compact G [the conjugate diagram of the function ~(X)]. It follows from the condition of completely regular growth that outside some set of discs S = {Sj},

Sj={X~C:

I~--ajl
(]=t,

2.... )

of zero linear densityt the uniform estimate

In I ~ (2) I = [h (art 2) + o

(i)l

I Xl.

(2)

holds. We s h a l l c o n s i d e r t h e d i s c s Sj t o be d i s j o i n t [5] and e a c h c o n t a i n i n g z e r o [2, p. 2 3 4 ] . The d i s c s S o b v i o u s l y s a t i s f y t h e f o l l o w i n g c o n d i t i o n :

at least

r.

limj - ~ Ia~I ~ ~0.

one (3)

LEMMA 1. L e t h ( O ) , 0 < 0 ~ 2w be a p e r i o d i c w i t h p e r i o d 2w and t r i g o n o m e t r i c a l l y convex function. Then t h e r e e x i s t s an e n t i r e f u n c t i o n w(X) w i t h t h e i n d i c a t o r h(O) whose z e r o s e t {bk} i s r e g u l a r . This function satisfies on t h e s e t S t h e a s y m p t o t i c e q u a l i t y l n l w ( k ) t = [ h ( a r g k ) + o ( 1 ) ] l X I. Proof.

F i x a number ~, 0 < B < 1.

N e x t , d e n o t e Xj ~ = sup I l l

and c o n s t r u c t

the discs

S T={2---C: [L--ail<~exp[--(L~)s], Then the set of discs S* + {Sj*} has zero linear density.

Indeed, the equalities

lim.-= Z,.,.<,. (", + exp [-- (2~)6)= '

'

= 1i'n"-~ [-'7- ( Z I , , j <~ " ) + -7- Zlail<~exp hold.

[-

(2*)q . ]

Since Zj* ~ =, we have t

exp [-- (;~'1t;l .< (~.*)= , Note that for every zero X ~ of the function r

holds.

J > ]o'

contained

in a disc Sj the inequality

Therefore ,

I

I

and we finally obtain

+

expI--(2>1:0

Thus, the set of discs S* has zero linear density. Now construct [6, p. 79] an entire function w(X) with the indicator h(O) whose zero {bk} lies outside the discs S* and is regular. Outside the discs I~ - bkl < exp(-Ibkl~0), ~ < D0 < i, for the function w(%) the uniform #A set of discs {i ~ C:

I~ - ~jl < rj} (j = i, 2 . . . . ) has zero linear density if

lira,. -~,(I/r),..i!, ~

28

jl <, "j =

O.

estimate in]w(X) I = [h(argl) + o(i)]I~ I holds (see [7, p. 239]). It now remains to note that for k > k 0 the discs I~ - b k l < exp(-Ibkl~0) are disjoint from the set of discs S. Lemma 1 is proved. We introduce the set ST of the locally analytic functions D(%) on the set S satisfying the condition B]:

slIp~Y:S exp [hi (arg ~) I t" I1] For

~ (~)~

we d e n o t e

THEOREM I.

by ~ j ( X )

the contraction

For any function

~ (g)~ ~ there

of ~(h)

by a c o m p o n e n t S j .

exists

a function

g(X)~

P satisfying

the

condition

g (X) - - ~i (X) = ~ ( X ) . q j ( X ) , X~S~ q~ (x) ~ ~ (sA. Proof.

Let

~ (~)~/~;

then one can find

(1 = 1 , 2 . . . . ),

j such that

[B (~) [
S.

Since D = U ~ Q G a , where Q is a convex region, there exists a convex region d ~ Q such that Gj & G + d E G(d). Let hd(arg%) = kd(-arg%), kd(argl) is the support function of the compact d. By Lemma i, there exists an entire function Wd(l) of exponential type that has a completely regular growth and on the set S satisfies the asymptotic equality

in I wa (~) t = [ha (arg ~) -~ o (l)l I ~ t. Consider the function N (~) = wd (~). 9

(~),

for the growth indicatrix of which the equality

h~ ( ~ g ~) = ha (arg ~) + h (arg ~) holds. Choose in each component Sj a closed contour s [surrounding zeros of the function ~(X) in Sj] so close to the boundary 3Sj that on the set s = Djs the equality

will hold.

Now consider the series ~.=

j

~j(z)N(~)

d.~,

(4)

where % ~ C ~ ~ j S j . Series (4) uniformly converges in the region C ~ ~ j g j , Indeed, for any compact K ~__C'\ ~jS~ the distance p(K, s between the component K and ~ is bounded below by some positive number 6. Therefore, the estimate

t

~j (z)

(5)

/V (2 i 7--- z) i ~ cl exp []9 (arg z) - - hx (arg z)] I z I ~ e~ exp (-- s o i z ])

holds, where g0 > 0, c 2 is a constann. Since there is at least one zero of the function ~(1) in each disc Sj, by (5) series (4) is majorized on the compact K by the convergent series

~'is c 3 exp (-- 3~01 a i l). Thus, series integral

(4)

is uniformly

convergent

in the region

C ~jjSj.

Note that

f o r any j t h e

j qT(z)N (>~) dz YJ (~) -- ,,: :v is)(t~-- ~) defines an entire function of I. This observation and the fact that the region C \~ 6jJj contains (see [6, p. 43]) annular regions {X: rj < l i i < rj+1}, rj + ~, imply uniform con-

29

convergence of series (4) on any compact of the complex plane. Consequently, series (4) defines an entire function g(X). The function g(l) belongs to P. Indeed, consider the discs

B s. = {s where 0 < ~ < i, X j e ~

z~lj.

Let first

Sj;

a point

Ixj~

X lie

[)~--ai I
Ill.

= inf

outside

exp (--[s

Estimate the quantity i/Iz - h i for

Rj and o u t s i d e

1 lz--Xl <expJX]~, Next, for the points X lying outside the disc we have

the disc

Ill

< Ixj~

s

then

z~l~.

IX - a j l

< i + rj and in the disc

IX] ~ IXj ~

i

Finally,

let

X~Ri, m~{X~C:

IXl -~ ~ I ksI o }(-]{k~-- C: ] ~ - - aj ! ~r~ q- 1};

then

z_:.Xl -~%exp[--l~.~l~l~.exp

i;v]l, I%1~ .

Since

j--~-]--i] -..~c,

J>Jo,

s

c=const,

we obtain 1

i=_~,1 .~exp[--cl&lq , Taking into account all the inequalities

Iz-Xl where c I is some constant.

X~Rj,

l>/o,

z~lj,

The last inequality and (5) give the estimate

~j(z)

z-~lj, ~ B j .

Ig(X)l ~ B e x p [hN ( a r g X ) e < hj, (arg X) for some obtained estimate of the plane. Therefore, g ( X ) ~ tion that Jp(V)(X k) = 0, residues one can directy fore

z~lj.

obtained above, we finally have

~exp(--c*l~]l~)'

~(z)(~ -z)

where A i s a c o n s t a n t ,

~,~o~Rj,

< A e x P ( - - c , l ;.l~)" exp ( - - % [ z l), 8 0 > 0 ' Substitute

(6)

in (4);

for the function

(6) g(X) we h a v e

+ g]-IXI, where e is a positive number so small that h N ( a r g X ) + j*. Since the set {Rj} has zero linear density (see Len~na i), the function g(X) can be extended from the set C\ UjRj over the whole P. Now consider the functions Jp(X). It follows from their definiX k ~ Sj, 0 g v g m k - I, j = p, and with the aid of the theory of ver"i f y that Jp (v) (X k ) = ~p ( v ) (Xk), Xk =~ Sp, 0 ~ v ~ m k - i. There-

g (~.) -

% (~.) = qa O,).q~, (~.),

where qp(~) are some functions holomorphic

in Sp.

~. ~

$1,,

p = t .....

Theorem I is proved.

Take in each disc Sj an arbitrary point si (j = I, 2, ..) and with the use of the functions h k(argX).lll define the matrix A = { jkY (k, j = i, 2 . . . . ), where

aj~. = exp [hi, (arg s./) j sjj J. Note that A is a Kothe matrix, LEMMA 2. The following D' k > 0 Such that

i.e., 0 ~ aj~: ~ a~+ l (i, k = I, 2 .... ).

inequalities

sup [hk (argO.) [Z I ] <

~ sj

30

hold.

For any integer k there exist D k > 0 and

]naj,~+ 1-c-D~: (j = 1, 2 . . . .

);

(7)

In a:~ ~< inf

[h~+1 (arg ~) ] ~ !] q- DI,.

(] = t, 2, . . .).

(8)

Proof. Prove the first inequality. Let sj* be a point in the closure of Sj at which the fu-~ction h k (argX).IX I attains its maximum on the set Sj. Condition (3) implies the relation limj_~Isj* - sjl/Isj] -- 0. Therefore limj_se [h~ (arg s*) - - h~ (arg sj)] = 0.

(9)

0

Since G k c Gk+1, for some positive zk the relation h k (argA) + z k < hk+ z (argX) (k = l, 2, ... ) holds. From this inequality and relation (9) we obtain hk (arg s3"*) ]s.*l < hk+l (argsj)]sj], j > J0. Therefore, one can find a number D k > 0 for which all j (7) will hold. Inequality (8) can be established in an analogous way. Lemma 2 is proved. Let us introduce several spaces necessary in what follows Let g = (Ej~ sequence of finite-dimensional Banach spaces. For 1 <- p g = we put

iV(A, g) = { x = (x~), x) ~ Ej: ~)=1 (ll xj ~) aj,n)P < .~,

Vm = 1.... j ;

X'~

Vm=l

sup(I[x~I]~a~m)<~,

~j)j=l ~ be a

. . . . };

.1

k~(A,g)

{x=(xj),x~E~:~,=,(ilx~lbaT~,n)P
a~=) <

~,

3.t =

.}"

1 . . . . }.

For the case Ej = C (j = l, 2, . .) we. p u. t E P ( A C) = s kp(A, C) ----k ~(A), I -~p ~ %1" oo. It is assumed thaL Xp(A, ~) and kP(A, g) are equipped with the natural topology. Since A is a Kothe matrix, the following two lemmas hold. L _ ~ _ 3 [4]. Let g * = (E4*, ~ ~ .*) be the sequence of strong duals of Ej. Then for 1 _< p < ~ the strong dual of XP{A, g) is3 isomorphic with kq(A, s *), where i/p + i/q = i. LEMMA 4 [4]. The following two assertions are equivalent, i. The space AZ(A, ?') is nuclear. 2. The space k=(A, ~ *) is nuclear. Moreover, if one of these assertions hold, then there exists a Hilbert norm J (lj on Ej such that for the sequence g = (Ej, ..!elJ) we have )~ (A, ~)--~X 3 (A, g) ~--)J (B) ~ % = ~)where the matrix B = (b~k) is defined by t condition bs = ajk for

3. On a Basis in the Space W. Let W • be the subspace orthogonal to W, i.e., the set of all the functionals in H*(D) vanishing on W. Denote M = {L(T)}, T ~ W i (L is the Laplace transform). The set M is uniquely determined by the zero set of the function ~(A); this means that any function ~ ( A ) ~ P for which the ratio ~(X)/r is an entire function belongs to M (see, e.g., [8]). Since H(D) is a Schwartz F-space, the strong dual of the factor space P/M is topologically identifiable (see [9]) with the space W according to the formula (/, ( L ( T ) + M)) = (/, T), Equality (i0) defines a mapping r W the notation [4]: H A, s is the equipped with the topology of simple set M with the topology induced from

T~H*(D),

/~-- W.

(10)

+ (P/M)*. For further description of P/M we introduce ring of germs of analytic functions at a point X convergence; M A is the ideal in H A generated by the HA;

(~I.16) ( t h e r e i s d e f i n e d i n Ej t h e n a t u r a l t o p o l o g y g e n e r a t e d by t h e t o p o l o g y i n H i ) ; H~(Sj) 1, 2, . . . ) a r e t h e Banach spaces of bounded a n a l y t i c f u n c t i o n s on Sj w i t h t h e norm sup

If(~)

!,/~H

(j =

~(&).

C o n s i d e r t h e m a p p i n g 0 j : H ~ ( S j ) + E{, a c t i n g by t h e r u l e 0 i ( f ) = [ f ] A + MA, A ~ S i n A, [ f ] A i s t h e germ o f a f u n c t i o n ~ H ~ S j ) at a p o i n t A. T h e ~ m a p p i n g 0i i s l i n e a r a~d continuous; moreover, from Weierstrass' t h e o r e m ( s e e [ t 0 , p. 2 9 ] ) i t s s u r ~ e c t i v i t y follows. Therefore, we c a n d e f i n e a norm on Ej p u t t i n g ~i !/!, =

iur {4 ' ~

/.=

H ~ (s~):

(i) =

.v}.

(ll)

ol

With this norm, Ej becomes a Banach space. Introduce the sequence [ = E j, ~ ~j) and consider the linear mapping p, defined on the set P, which assigns to each ~unction ~ (~) ~ P the sequence (pj(~ISj)) (j = i, 2 . . . . ). LEMMA 5.

The mapping p carries the space P into the whole k~(A, g).

Proof. First prove the inclusion p c k~(A, m we have ~(~) ~ Pm, i.e.,

@).

If ~(l) ~ P, then for some integer

I W (k) t --< I1 ~F Ib .exp [h~ (arg X) t XI1. From t h e l a s t

estimate,

Lemma 2 [ i n e q u a l i t y

(7)],

and d e f i n i t i o n

(11) we o b t a i n

IIp~(W I Si) Ili ~< ll ~P'IS~ ll.=(s~) ~< [] T lie,,, exp [sup hm (arg s s II < sj

~< U~F lip,, exp [hm+~(arg sJ]

sll

+

D,,]

=

I] R* II*',, .a~,,~+~. exp Dm.

Consequently, p (W)--~_k = (A, @) and, as the last inequalities show, the mapping O is continuous. Now prove the surjectivity of 0. Let x----(xj)j~=1~ k ~ (A, ~). Then there exists an integer m such that Q - sup ]xj ~jajm -I < -, o r ! x j ] j ~ Q~lim (J = I, 2 . . . . ). Therefore [see definition (ll)] there can be found Oj(l)~-= ~I~[Sj), ~or which p j ( ~ j ) = xj and 1! "qj 1]HO~(sp << 2Qaj.~

(] = I, 2 . . . . ).

By Lemma 2 [inequalities (8)], for nj(~) the estimate

I ~s (~) I -.<2Q exp [h,,~ (arg Ss). I Ss I1 ~ 2Q exp [h,,+~ (arg X) t X I t exp D;,,

~ ~ S~,

h o l d s , i . e . , q(~) = { q j ( a ) } ~ gg. But t h e n by Theorem 1 t h e r e e x i s t s a f u n c t i o n g ( ~ ) ~ P s a t i s f y i n g t h e c o n d i t i o n g (~) -- ~]s (s = Op (s (s (] = i, 2 . . . . ), s ~ Sj, qs (s ~ H (Ss). This means t h a t 0(g) = x. Lemma 5 i s p r o v e d . Thus, t h e mapping O: P + k~(A, ~) i s c o n t i n u o u s and s u r j e c t i v e . I t r e m a i n s t o remark t h a t M = k e r 0- C o n s e q u e n t l y , by t h e open mapping t h e o r e m f o r L F - s p a c e s [11] P/M and k~(A, g) a r e t o p o l o g i c a l l y i s o m o r p h i c . S i n c e t h e s p a c e P, and t h e r e f o r e a l s o P/M, i s n u c l e a r [12, pp. I16 and 151], k'(A, g) is also a nuclear space. THEOREM 2. There exists in the space W a Schauder basis the elements of which can be divided into groups o{fjp} (p = 1 . . . . . dimEj) (j = i, .. ) each respectively belonging to the linear envelope the functions {zkexp(IrZ)}, I r ~ S ', 0 <- k g m r - i. Proof. Define in each Ej the norm I ~ which is determined by the inner product (see Lemma-~).-]-- Denote ~' = (Ej, I ]j)" Then, using the nuclearity of k~(A, @), by Lemma 3 we obtain k ~ (A, @ ) - - ~ (A,~)(i.e., these spaces are topologically equivalent). In each Ej construct the basis (see [13])

y~r = (Srp " [(=--)vv k! )~ ] s

~ S],

if r = p, and 5rp = 0 for r ~ p. S, then

(go

s ~ $1,

}O~.r'"{-:1[}'r/

0 ~ k ~ mr - - i, 6r,,

=

l,

If we now denote epn = zPexp (In.Z), 0 ~ p ~ m n

(%.) [g + M]) = fa( ~ ~ ) = ~ ' ) (7~,,),

"2" (~.) ~

i, l n ~

P,

and, identifying P/M with Imp, we obtain (00 (%,~), Y~r) = 5 n r ' ~ p ~ O .~ k .~< m r

i,

0 /

(n, r = ~ . . . . /

),

i.

The latter means that as the dual space of (Ej, I lj) we can take the set Fj = {linear envelope of ~(epn), 0 <_ p <- m n - i, X n ~ S j } , in which the dual norm I lj* is introduced. So Fj is a finite-dimensional Hilbert space. Denoting .g = (Fj, I ljr By Lemmas 3 and 4 we obtain

(12)

32

It is known [12] that the system 6m = (0 ..... 0, i, 0 .... ) constitutes a Schauder basis in XZ(B). Consequently, the space (P/M)* also has a Schauder basis, which can be obtained as a union of orthonormal (in the norm I lj*) bases of each Fj. Since ~ is a topological isomorphism of the spaces (P/M)* and W, there is a Schauder basis defined in W which, as follows from the description [4] of the isomorphisms in (12), possesses the properties indicated in this theorem. Theorem 2 is proved. LITERATURE CITED i. 2. 3. 4.

5. 6. 7.

Vo V. Napalkov, Convolution Equations in Multidimensional Spaces [in Russian], Nauka, Moscow (1982). B. Ya. Levin, Distribution of Zeros of Entire Functions [in Russian], Gostekhizdat, Moscow (1956). A. F. Leont'ev, Sequences of Polynomials of Exponentials [in Russian], Nauka, Moscow (1980). R. Meise, "Sequence space representations for (DFN)-algebras of entire functions modulo closed ideals," Journal fur die Reine und Angewandte Mathematick, 363, 59-95 (i985). Io Fo Krasichkov-Ternovskii, "A geometric lemma useful in the theory of entire functions and Levinson-type theorems," Mat. Zametki, 24, No. 4, 531-546 (1978). A. F. Leont'ev, Series of Exponentials [in Russian], Nauka, Moscow (1976). A. Fo Leont'ev, Generalizations of Series of Exponentials [in Russian], Nauka, Moscow

(1981). 8. 9. I0. iio

12. 13.

!~ F. Krasichkov-Ternovskii, "Invariant subspaces of analytic functions. I. Spectral synthesis on convex regions," Mat. Sb., 87, (129), No. 4, 459-489 (1972). A. Grothendieck, "Sur les espaces (F) et (DF)," Summa Bras. Math., 57-123 (1954). L. Hormander, An Introduction to Complex Analysis in Several Variables, Elsevier, New York (1973). J. Dieudonne and L. Schwartz, "Duality in the spaces (F) and (LF),"Matematika [a collection of articles in Russian], ~, No. 2, 77-117 (1958). The probable French original: "La dualite dans les espaces (F) et (LF)," Ann. Inst. Fourier Grenoble i, 61-101 (19491950). A. Pietsch, Nuclear Locally Convex Spaces, Springer-Verlag, New York (1972). R. Meise, Ko Schwerdtfeger, and B. A. Taylor, "On kernels of slowly decreasing convolution operators," Preprint, 32 (1985).

33