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The Summability of Fourier Series
Published online by Cambridge University Press: 03 November 2016
Extract
One of the oldest problems in the theory of Fourier series is that of looking for a criterion that a Fourier series shall converge. No one, however, has been able to find a simple, necessary and sufficient condition for this. Thus, for instance, bounded variation of the function is sufficient but not necessary. Continuity is neither necessary nor sufficient. That is to say, there are functions whose Fourier series converge at points of discontinuity, and others whose Fourier series diverge at points of continuity. If we consider the same problem for Cesàro summability of any particular order, similar difficulties arise.
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- Copyright © Mathematical Association 1931
References
* A paper read at the Bristol Meeting of the British Association, September 1930.
† Math. Zeitschrift, vol. 19.
‡ See papers by Hardy and Littlewood, Paley, Verblunsky, and the writer in the Proceedings of the Cambridge Phil. Soc. and of the London Math. Soc.