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A Convergence Theorem for Double L 2 Fourier Series

Published online by Cambridge University Press:  20 November 2018

Richard P. Gosselin
Richard P. Gosselin*
Affiliation:
University of Connecticut
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Our aim in this paper is to extend a known theorem about the convergence of subsequences of the partial sums of the Fourier series in one variable of class L 2 to Fourier series in two variables of the same class, (1, p. 396). The theorem asserts that for each function ƒ in L 2, there is a sequence {m V } of positive integers of upper density one such that

Smv(X;ƒ)

converges to ƒ almost everywhere where sm(x;f) denotes the mth partial sum of the Fourier series of ƒ.

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 10 , 1958 , pp. 392 - 398
Copyright
Copyright © Canadian Mathematical Society 1958

References

1.Gosselin, R. P., On the convergence of Fourier series of functions in an Lp class, Proc. Amer. Math. Soc, 7 (1956), 392-397.Google Scholar
2. Sunouchi, G., Notes on Fourier analysis XXXIX, Tohuku Math. J.(2), 3 (1951), 71-88.Google Scholar
3.Zygmund, A., Trigonometrical series (Warsaw, 1935).Google Scholar