No CrossRef data available.
Article contents
The Absolute Summability of Fourier Series*
Published online by Cambridge University Press: 03 November 2016
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
- Copyright
- Copyright © The Mathematical Association 1933
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.