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The Expression of Trigonometrical Series in Fourier Form

Published online by Cambridge University Press:  20 November 2018

George Cross
George Cross*
Affiliation:
University of Western Ontario
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In a paper published in 1936 Burkill (2) proved that, if the trigonometrical series

1.1

is bounded except on a countable set and if the series obtained by integrating series (1.1) once converges everywhere, then the coefficients can be written in Fourier form using the C1P-integral. In §3 of this paper an analogous result is shown to be true when (1.1) is bounded (C, k), k < 0. The proof of this depends on generalizations of theorems by Verblunsky and Zygmund and both of these generalizations are obtained in §2.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 12 , 1960 , pp. 694 - 698
Copyright
Copyright © Canadian Mathematical Society 1960

References

1. Burkill, J.C., The Cesàro-Perron scale of integration, Proc. London Math. Soc. (2), 39 (1935), 541–552.Google Scholar
2. Burkill, J.C., The expression of trigonometrical series in Fourier form, J. London Math. Soc, 11 (1936), 43–48.Google Scholar
3. Hardy, G.H., Divergent series (Oxford, 1949).Google Scholar
4. Saks, S., Theory of the integral (Warsaw, 1937).Google Scholar
5. Verblunsky, S., On the theory of trigonometric series VII, Fund. Math., 23 (1934), 193–236.Google Scholar
6. Zygmund, A., Trigonometric series (2nd éd.; Cambridge, 1958). II.Google Scholar