On the Summability of Fourier Series

MATHEMATICS: HILLE AND rAMARKIN

VoL. 14, 1928

915

ON THE S UMMA BILIT Y OF FO URIER SERIES By EINAR HILLS AND J. D. TAMARw PRMIcKroN UNIVERSITY AND BROWN UNIVERSITY Communicated November 7, 1928

1. The present note is intended as a contribution to the question: What conditions must be satisfied by a method of summation in order that it shall sum the Fourier series of an integrable function almost everywhere to the function? Certain methods of summation are known to have this property, e.g., the arithmetic means (C, k) with k > 0, the method of de la Vallee-Poussin, the method of Abel-Poisson (and analogous methods). Others do not have it, the methods of Borel and of Euler being the most noteworthy ones. All these methods are simple and natural definitions of summability, but after all merely special instances of possible definitions. As a preparation for an attack on the general problem formulated above we have investigated two classes of methods of summation from this point of view, each class having a fair degree of generality and closure and, furthermore, known to contain a continuum of effective methods. We are going to be concerned with (a) the method of Norlund, and (b) the method of Hausdorff-Hurwitz-Silvermann. 2. The method of Norlund is defined in the following manner:' Let So, Si, .., ,S, . .. be the partial sums of the given series, convergent or not, and put Tn

=

(poSn + p1Sn.1 +

...

+ PSo) F

F Pt

=

P"'

(1)

0.

where I P1 I is a sequence of given positive numbers with the sole restriction that pnPn1-* 0. If T, - T we say that the series is summable (N, p,,) with the sum T. Let f(x) be integrable Lebesgue in (-x, ir) and let ° + j (am cos mx + bm sin mx)

(2)

be its Fourier series. Applying N6rlund's method of summation to the partial sums of (2) we are led to the following expression for T.:

Nn[f(x),lpn]

=

f f(t+x) [P/2+

P#,#cOS mt]dt (3)

For the present we restrict ourselves to a mentioning of the following three cases.

916

MA THEMA TICS: HILLE AND TAMARKIN

PRoc. N. A. S.

I. If

(r + 1) IPpn - r - Pn-r-j1
= 0)

(4)

for every value of n, the series (2) is summable (N, pn) to the sum f(x) almost everywhere. This is a consequence of the Fej&-Lebesgue theorem2 together with the Lemma: Any series which is summable (C, 1) = (N, 1) is also summable (N, Pn) if condition (4) is satisfied for every n. Condition (4) is satisfied, e.g., if either P. . Pn + 1 when n _ no, or Pn _ pn + I when n > no and lim p,, > 0. II. Suppose P. > Pn + 1, Pn - 0 and P,, -* . Putting

p(z)

=

E P.z 0

we find that the boundedness of

P-1 _

|d

[p(?i)t-'] I dt(5

is sufficient to ensure summability (N, P,) of (1) to the sum f(x) almost everywhere. It could be observed that the arithmetic means (C, k) satisfy condition (4) if k > 1, and condition (5) if 0 < k < 1. III. The choice pn = (n + 1)-1 is consistent with the assumptions under II. The corresponding means are known as the harmonic means;3 they are weaker than any arithmetic means of positive order. We find that the expression (5) is not bounded in this case and the Lebesgue constants corresponding to this definition are actually O(log n). It follows that there exist continuous functions whose Fourier series are not summable (N, (n + 1) 1) almost everywhere.4 A necessary and sufficient condition in order that

>f(x)

N. [f(x), n + 1] is that

dt flog (n + 1)]-1 so(t) sin (n + 1)t log 1 -t -

1 nf

where

we(t)

=

f(x + t) + f(x

-

t)

-

2f(x).

0,

(6)

VOL. 14, 1928

MA THEMA TICS: HILLE A ND TA MARKIN

A sufficient condition is given by F 1 11fa p(t + )-f(t) lim [log -i]

Ilog 1t-dt =0.

917

(7)

In both formulas 5 denotes a fixed small positive number. In a general way we may expect a N6rlund transformation to become less powerful the faster p, tends to zero. Necessary and sufficient conditions for equivalence or relative inclusion to hold between two Norlund transformations, have been found by M. Riesz.5 With the aid of these conditions we can prove that if an arbitrary series is summable (N, pn)

where Ep, < po, then it is necessarily also summable (N, + There exists, consequently, a continuum of (N, P,) means which are ineffective with respect to Fourier series. 3. Hurwitz and Silverman gave -a characterization of the regular anmSm such that the matix (anm) is permutable transformations Tn = 0

with the matrix corresponding to (C, 1).6 Later but independently the problem was attacked by Hausdorff7 who showed that to every such transformation there corresponds a function q(u) which is of bounded variation on (0, 1) with q(O) = 0 and q(l) = 1 and such that T

=

E Sm (nu)fum(1

-

u)" - mdq (u)

(8)

T the series whose partial sums are the Sn is said to be summable (H, q(u)) to the sum T. In the case of a Fourier series Tn becomes

If Tn -

H.[f (x),qJ

= f

r+

f(t) Hn(t

-

x)dt

(9)

where

-i) H.(v) =-a[(1 7ro

(1 -u +

ueiv)ndq(u)I

(10)

The series (1) is summable (H, q) to the sum f(x) almost everywhere if q(u) possesses a derivative which is of bounded variation in (0, 1). The same result is true if

q'(u)

=(1 - u)1qi(u) where O

and

< k < 1

(1-u)- 1 [ql(l) - ql(u)]

is of bounded variation in (0, 1). The case ql(u) = k corresponds to the

Ces.ro means (C, k).

918

MA THEMA TICS: G. A. MILLER

PROC. N. A. S.

A simple example of a Hausdorff transformation which fails to sum Fourier series almost everywhere is obtained by taking q(u) = 0 when 0 < u < 1/a and = 1 when 1/a _ u . 1. This choice leads to Euler's method Ep with p = log a/log 2 using Knopp's terminology.8 The corresponding Lebesgue constants are actually O(log n). 1 N. E. Norlund, Lunds Universitets Arsskrift, N. F., avd. 2, 16, 3, 10 pp., 1919. 2 L. Fejer, Math. Annalen, 58, 51-69, 1904. H. Lebesgue, Math. Annalen, 61, 251-260, 1905, and Annales Sci. Fac. de Toulouse, (3) 1, 25-117, 1909, especially pp. 88-90. 3 M. Riesz, Proc. London Math. Soc., (2) 22, 412-419, 1923. 4 Professor M. Riesz has kindly informed us that this result has been known to him for some time. A proof has been worked out by one of his pupils, Mr. N. K. A. Juringius, but has not been published. 5 Loc. cit., p. 413-414. 6 W. A. Hurwitz and L. L. Silverman, Trans. Amer. Math. Soc., 18, 1-20, 1917. IF. Hausdorff, Math. Z., 9, 74-109, 280-299, 1921. 8 See C. N. Moore, these PROCgi3DINGS, 11, 284-287, 1925. 0

GROUPS IN VOL VING A CYCLIC, A DIC YCLIC, OR A DIHEDRAL GROUP AS AN INVARIANT SUBGROUP OF PRIME INDEX By G. A. MILLB3R DEPARTM8NT OF MATHZMATICS, UNIVERSITY OF ILLINOIS Communicated November 1, 1928

A general method for constructing all the possible groups which involve a given group H as an invariant subgroup of index p, p being any prime number, was developed in a recent number of these PROCE8DINGS,' and several illustrative examples which are special cases of the groups denoted by the heading of the present article were -then given. Since these groups constitute elementary infinite categories of groups of finite order it seemed desirable to determine them completely by the given method and to endeavor to express the results in a convenient form, especially since the general theory of determining all the possible groups containing a given invariant subgroup has not yet received much attention. For the sake of determining all the groups in question it is desirable to add to the general method to which we referred the following theorem: If 5i transforms an operator of the abelian constituent group H1 into its kth power then the group to which SlS2... spt gives rise is conjugate with the one obtained by using for so the kP - 1 . ..... + k + 1 power of the given operator. From this theorem it results directly that if H is the cyclic group of order pm, m > 2 when p = 2 and m > 1 for other values of p, and if S1S2 * * * Sp 1 power transforms the operators of this cyclic group into their pm + then there is one and only one such group of order pm 1, since we obtain