Convergence Criterion For Fourier Series Research Articles

A generalization of lebesgue’s convergence criterion for fourier series

For generalized Dirichlet integrals of type \(\int_0^1\ f(x)(e^{irx}-e^{airx}){dx\over x}\) a somewhat sharpened version of Lebesgue’s well-known convergence criterion for Fourier series is proven. Integrals of this type are of importance in the general theory of eigenfunction expansions inaugurated by investigations of Birkhoff, Tamarkin, and Stone.

Convergence criteria for Fourier series

The following convergence criterion of Fourier series is due to M. Izumi, S. Izumi and the author:THEOREM. Let Δ ≥ 1. If(i) , and(ii) as t → 0for an a, 0 < a < 1 and for a δ, 0 < δ < π, then the Fourier series of φ(t) is convergent at the origin.The object of this paper is to generalize the above theorem in the Hardy-Littlewood direction.

A convergence criterion for Fourier series

for a special xo does not imply convergence of the Fourier series S[f] of f(x) for that xo. The Hardy-Littlewood convergence test [4, p. 63] gives the convergence of S [f] at xo, provided that the condition (1) is satisfied, and the coefficients of Slf], a,, and bn are O(n-0) for some positive 0. The aim of the note is to give a sufficient condition on the function w(x0, h) which will insure the convergence of the S[f] at the point xo. A similar condition was given by the author [3] under some additional assumptions. The first object of this note is to give a direct proof for this result. Following A. Zygmund [4, p. 186] we shall use the following definition: A positive function L(t) defined for 0 0, L(t)t6 is ultimately a decreasing and L(t)t-' increasing function of t for t->O. From this definition we deduce immediately: If L(t) is slowly varying, then

On convergence criteria for Fourier series, II

On convergence criteria for Fourier series, II

On convergence criteria for Fourier series, I

On convergence criteria for Fourier series, I

A new convergence criterion for Fourier series

A new convergence criterion for Fourier series

Convergence criteria for Fourier series

Convergence criteria for Fourier series

Notes on Fourier analysis (XLVI): A convergence criterion for Fourier series

Notes on Fourier analysis (XLVI): A convergence criterion for Fourier series

A Note on Cesaro Summability of Fourier Series

1. It is known that the (C, a) continuity of a function is not sufficient for the (C, a) sumnmability of its Fourier series at a point. Many sufficient conditions for (C, a) summability of a Fourier series were found by many authors.' The object of this note is also to find a sufficient condition which arises from a convergence criterion for Fourier series recently proved by the author.2 Let +(t) be an even integrable periodic function with period 2ir and its Fourier series be

Notes on the Theory of Series (XVII) : Some New Convergence Criteria for Fourier Series

Journal of the London Mathematical SocietyVolume s1-7, Issue 4 p. 252-256 Notes and papers Notes on the Theory of Series (XVII) : Some New Convergence Criteria for Fourier Series G. H. Hardy, G. H. HardySearch for more papers by this authorJ. E. Littlewood, J. E. LittlewoodSearch for more papers by this author G. H. Hardy, G. H. HardySearch for more papers by this authorJ. E. Littlewood, J. E. LittlewoodSearch for more papers by this author First published: October 1932 https://doi.org/10.1112/jlms/s1-7.4.252Citations: 16AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Citing Literature Volumes1-7, Issue4October 1932Pages 252-256 RelatedInformation

Note on a Certain Power Series

* Received December 13, 1929. t Reference to this result has been made by G. H. Hardy and J. E. Littlewood, A convergence criterion for Fourier series, Math. Zeitschrift, 28 (1928), pp. 612-634 (p. 633 footnote 32)), and by Einar Hille, Note on a power series considered by Hardy and Littlewood, Journal London Math. Soc., 4 (1929), pp. 176-182 (pp. 181-182). There is a slip in Hardy and Littlewood's statement of the result, which is corrected by Hille. The condition r = o on p. 182 of Hille's paper is presumably a misprint for y>'O. Actually, however, I had considered the series (1) only for O<y < 1, and I confine myself to this case here, this being the only one which is of interest for the application in view. The same result has been obtained independently and by a different method by Hille in a paper as yet unpublished but kindly placed at my disposal by the author. The present note is written at the request of Professor Rille. 241 18

A convergence criterion for Fourier series

for a special xo does not imply convergence of the Fourier series S[f] of f(x) for that xo. The Hardy-Littlewood convergence test [4, p. 63] gives the convergence of S [f] at xo, provided that the condition (1) is satisfied, and the coefficients of Slf], a,, and bn are O(n-0) for some positive 0. The aim of the note is to give a sufficient condition on the function w(x0, h) which will insure the convergence of the S[f] at the point xo. A similar condition was given by the author [3] under some additional assumptions. The first object of this note is to give a direct proof for this result. Following A. Zygmund [4, p. 186] we shall use the following definition: A positive function L(t) defined for 0 0, L(t)t6 is ultimately a decreasing and L(t)t-' increasing function of t for t->O. From this definition we deduce immediately: If L(t) is slowly varying, then

Notes on the Theory of Series (VII): On Young's Convergence Criterion for Fourier Series

Proceedings of the London Mathematical SocietyVolume s2-28, Issue 1 p. 301-311 Papers Notes on the Theory of Series (VII): On Young's Convergence Criterion for Fourier Series G. H. Hardy, G. H. HardySearch for more papers by this authorJ. E. Littlewood, J. E. LittlewoodSearch for more papers by this author G. H. Hardy, G. H. HardySearch for more papers by this authorJ. E. Littlewood, J. E. LittlewoodSearch for more papers by this author First published: 1928 https://doi.org/10.1112/plms/s2-28.1.301Citations: 12AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Citing Literature Volumes2-28, Issue11928Pages 301-311 RelatedInformation