On the Summability of Fourier Series. Fourth Note

376

MATHEMATICS: HILLE AND TAMARKIN

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. 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).