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PROC. N. A. S.
there is an application to topological differential geometry which we shall consider elsewhere. 1 E. Kasner, "The Trajectories of Dynamics," Trans. Amer. Math. Soc., 7, 401-424 (1906). Also "Differential-Geometric Aspects of Dynamics," Princeton Colloquium Lectures on Mathematics (1913), especially Chap. I and Chap. III, where some of the properties are stated in projective language. 2 Bull. Amer. Math. Soc., 14, 356 (1908), where the results are stated. Also Bull., 36, 51 (1930). We assume that the field in (1) is not null, and that the surface in (2) is not a plane; these degenerate cases give merely the o 2 straight lines and so correspond. 3 The term linear is here used in its general or, topological, rather than its more usual projective or algebraic meaning. See author's St. Louis address, Bull. Amer. Math. Soc., 11, 307 (1905).
ON THE SUMMABILITY OF FOURIER SERIES. FOURTH NOTE By EINAR HILLE AND J. D. TAMARKIN PRINCETON UNIVERSITY AND BROWN UNIVERSITY
Communicated April 23, 1931
1. In the present note we continue the discussion of the summability of Fourier series by the method [H, q(u) ] of Hurwitz-Silverman-Hausdorff.1 We extend the discussion to the conjugate of a Fourier series and also to the derived of the conjugate of Fourier series of functions of bounded variation. A comparison between the results for Fourier series and for their conjugate series is rather striking. We use the notation of our third note except for the following modifications. We put 1 7y(u) = q(u) -u, O u <1
C(M)
=
cos
J(e) =
{u-y(u)du,J S(t) = J'sin (u'y(u)du,
fQ(u)
fi+d
Q(1)
-
Q(u)
du,
(2) (3)
where Q(u) is the total variation of q(v) on 0 _ v . u. The conjugate of the Fourier series of a function f(x) C L is co
(4)
E (-cos nx + an sin nx), n=i
where ao
f(x)1 aO
2
c=
a
+
n=l
(a,
+
bn sin
ff(t)e
dt,
cos nx
nx).
(5)
On setting -
ib5 =
(6)
MATHEMATICS: HILLE AND TAMARKIN
VOL. 17, 1931
377
and denoting the partial sums of the series (4) and (5) respectively, by $,(x) and s"(x), we have
an(X)
=
sn(X) + isn(X)
=
22
+ v=1 Zc,e .
(7)
Assuming f(x) to be real we obtain for the nth [H, q] mean of the sequence { o-,(x) } the expression2
Vntf(x), q] =
rrf(x + t)!§n(t)dt = Hn[f(x), q] + iHn[f(x), q]
(8)
where
'tO"(t)
(1 - u+
cot 27 2 -
ue-i)ndq(u) -i]
u + ue7s)ndq(u) = Hn(t) + iHn(t).
+ 2uJ (1
(9)
A point x will be called (F) regular or (L) regular with respect to f(x) if (i)x is a point of continuity of f(x) or (i')
J'If(x + T) - f(x - r) dr = 0(),
(10)
while, in both cases, (ii) the limit ow
f (x--)
1
lim
t
(r
ULf(x + t) -f(x - t)] cot dt
(11)
exists and is finite. The points which were designated in the third note as regular or pseudo-regular will be called here (F) regular or (L) regular. It is well known that almost all points, and, in particular, all (Fv) regular points are (L) regular. The nth [H, q] mean of the conjugate series (4) is
HnI[f(x), q] = f[f(x + t)
-
f(x
-
t)]"Hn(t) dt.
(12)
The method [H, q] will be said to be (F) effective or (L) effective if the series (4) is summable [H, q] to the sum ,7(x) for every f(x) CL at every (F) regular or at every (L) regular point of f(x), respectively. The derived series require the following notation and terminology. We assume now f(x) to be periodic and of bounded variation on (- 7r, 7r). A point x will be called (L') regular with respect to f(x) if (i) f'(x) exists and (ii)3
J'Idlk(T)I = o(t)
(13)
378
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PROC. N. A. S.
where
(t) = f(x + t) - f(x - t) -2t f'(x).
(14)
Similarly, a point x is defined to be (L') regular with respect to f(x) if (i) the limit rlimn
(t)[sin -2dt=f'(x)
(15)
exists and is finite and (ii)
J pId(r) = o(t)
(16)
where
(17)
It is well known4 that, except for sets of measure zero, all points are (L') regular and (L') regular. The nth [H, q] mean of the derived sequence { ¢X(x) } is
£njf(x), q] = f.tn(t) d1f(x + t) = Hn,[f(x), q] + i1'[f(x), q] (18) where H. [f(x), q] and H,,n[f(x), q] are the nth [H, q] means of the series (5) and of its conjugate series (4), respectively. It is found that
H, [f(x), q]
-
f'(x) = J Hn(t)dP(t)2
(19)
rf
H,'[f(x), q] = f HIn(t)dyp(t).
(20)
The method [H, q] will be said to be (L') effective or (L') effective if it sums the derived of the series (5) to the sum f'(x) and the derived of the series (4) to the sum7"'(x) at all the points x which are (L') or (L') regular, respectively, f(x) being any periodic function of bounded variation on
(-Tr, 7r).
2. Our results for the conjugate series are stated in Theorem 1 below. For the sake of comparison we include the corresponding results for the ordinary Fourier series, of which a part was proved in our third note. The symbol C L used in this section refers to the infinite interval (- co, co). It is assumed that q(u) satisfies the necessary and sufficient conditions of Hausdorff for the regularity of [H, q]. In addition we assume that lim J(e) is finite,
where J(e) is defined by (3).
(21)
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MA THEMA TICS: HILLE AND TAMARKIN
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THEOREM 1. A sufficient condition for an [H, q] satisfying (21) to be (F) effective is that (F) effective is that (22.2) L.
C(Q) cL.
S(Q)
(22.1)
n
A necessary condition for an [H, q] satisfying (21) to be (L) effective is that (L) effective is that X-_1/2
lim n /
n-o on-1
p(t) C(nt) dt
=
(.1n/2
(23.1) limonj" /o {(t)-S(nt) dt = 0 (23.2) 1 n-->
02
which is equivalent to {-n 1/2
(24.1) rin
lim n2 fp:(t) C'(nt) dt = O, n-+. Jn where
V10(t)
=
f(x + t)
-
f(x
-
t
ntdt
0,(n4-.1/S
lim n2J i(t)Sl(nt)dt = O, (24.2) (24. 1) n--0 co
-
t), (pi(t) =J'p(Tr)dr, 4/,l(t) =JPo(r)dr. (25)
A sufficient condition for an [H, q] satisfying (21) to be (L) effective is that (L) effective is that
tjC(Q)| < AQ())
tIS(Q)I| AQ(),
(26.1)
(26.2)
where A (t) is used as a generic notation to designate a function which is non-negative and of bounded variation on (1, co) and such that the integral ,o A (t) dt (27) Ea
is finite. The results for the derived series are as follows: THEOREM 2. A necessary condition for an [H, q] satisfying (21) to be
(L') effective is that -1/s
(L') effective is that ,X
C(nt)d/'(t)
lim fn n-.-G
1/
n
= 0,
which is equivalent to n-1/2 lim n2 / C'(nt)/.(t)dt .
co
(28.1)
1
-Jn-1
lim n n")Inco 1
rn= 0.
(29.1) lim n2 co
S(nt)d(t)= 0, (28.2) /2
S'(nt)(p(t)dt = 0. (29.2)
n-1
A sufficient condition for an [H, q] satisfying (21) to be (L') or (iL') effective is (26.1) or (26.2), respectively, with the additional assumption that A (t) be continuous on (1, co). A necessary condition for a regular [H, q] to be (F) effective is simply
C(Q)
a L,
(30.1)
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MATHEMA TICS: HILLE AND TAMARKIN
PROC. N. A. S.
as it has been proved in the third note. A natural question, whether
S(t)
cL (30.2) is necessary for a regular [H, q] to be (F) effective, remains open. It would be of some interest also to exhibit a regular [H, q] which is (F) effective without being (F) effective or vice-versa. We have reasons to believe that such a construction is possible. 3. Sufficient conditions for effectiveness of a different type are available on the basis of the discussion in our second note. We extend the definition of q(u) outside of (0, 1) by setting q(u) = 0 for u < 0 and q(u) = 1 for u > 1. We set
QQ(h) = maxfdu[Q(u + t) - Q(u)]|
(31)
and assume that J;
(h)
is finite.
(32)
We have then THEOREM 3. Condition (32) is sufficient in order that a regular [H, q] be (F), (F), (L), (L), (L') and (L') effective. Condition (32) requires that 1Q(h) - 0 as h -*0, which implies the absolute continuity of q(u).5 A particularly simple and important case is that in which q(u) is a convex function satisfying (32). The corresponding methods [H, q] which were mentioned in our second note appear to be equivalent, at least in so far as Fourier series are concerned, to a class of methods of summation studied by F. Nevanlinna from the point of view of (F) effectiveness.6 1 Cf. these PROCEEDINGS 14, 915-918 (1928); 15, 41-42 (1929); and 16, 594-598 (1930). 2 Hn(t) is the kernel of our earlier notes defined by the formulas (10) of the first and (2) of the third note. Both contain a misprint; the first exponental should be replaced by its conjugate. The third note contains a number of misprints. An annoying one occurs in formula (10) which should read p(t)/tCL. The statement of Theorem 4 is incomplete; it is necessary to assume that g(u) vanishes outside of a finite interval or satisfies some similar additional condition at infinity. 3 The notation here differs from that of the third note. 4 A. Plessner, Mitt. Math. Seminars Univ. Giessen, Heft 10 (1923).
6 A. Plessner, J. Math., 160, 26-32 (1929). 6 Oversikt Finska Vet.-Soc. Forhandl., 64, A, No. 3, 14 pp. (1921-1922).
