Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-14T07:29:28.124Z Has data issue: false hasContentIssue false
  • English
  • Français

The Absolute Summability of Fourier Series*

Published online by Cambridge University Press:  03 November 2016

L.S. Bosanquet
Get access Rights & Permissions [Opens in a new window]

Extract

The problem under discussion is the relation between the behaviour of a function ɸ (t) near a particular point t = x and the behaviour of the partial sums sn of its Fourier series at that point. We start from Hardy and Littlewood’s theorem † that if ɸ(t) tends to a limit in the Ceaáro sense as t → x, then the Fourier series at that point is summable in the Cesàro sense, and conversely. The exact order of the Cesàro means in each case is unspecified.

Type
Research Article
Information
The Mathematical Gazette , Volume 17 , Issue 226 , December 1933 , pp. 300 - 302
Copyright
Copyright © The Mathematical Association 1933 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

*

A paper read to the British Association, 1933.

References

page no 300 note † Math. Zeitschrift, 19 (1924), 67-96. This theorem has been extended and refined by several writers. For references to relevant papers see a paper by the present writer shortly to appear in the Proc. London Maths.soc.

page no 300 ‡ We shall suppose for simplicity that ɸ(t) is integrable L and even, that its Fourier series is cos nt and that x =O.

page no 300 § Actually for K > α

page no 300 ǁ Actually for K > α + 1.

page no 301 * The results above were obtained in collaboration with Dr. A.C. Offord, and some of them will be published in Compositio MathernaticaCompositio Mathernatica.

page no 301 † Proc. Edinburgh Math. Soc. (2), 2 (1930-31), 1-5. The definition of absolute summability (A) given here is equivalent to Whittaker‘s.

page no 301 ‡ Ibid.,’_129–134, and Proc. London Math. Soc. (2), 35 (1933), 407-424.

page no 302 * These results will be published in the Proc. Edilzburgh Math. Soc.

page no 302 †; The series would then be said by Fekete to be absolutely summable (C, K ) . Fekete, M. , Math. és. Termsrész.Ért (1911), 719–726.Google Scholar Kogbetliantz, E. , Bull. des Sci. Math. (2), 49 (1925), 234–256.Google Scholar A series which is absolutely summable (C) is also absolutely summable (A). Fekete, M., Proc. Edinburgh Math. Soc. (2), 3 (1932),132–131.CrossRefGoogle Scholar

page no 302 ‡ So far I have only considered the cases where a is an integer.

page no 302 § Compare Hardy and Littlewood, loc. cit., and Bosanquet, , Annals of Math. (2), 33 (1932), 758–77.CrossRefGoogle Scholar

page no 302 ǁ Compare a paper by Offord to be published in the Proc. London Math. Soc.