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A Convergence Theorem for Double L2 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 L2 to Fourier series in two variables of the same class, (1, p. 396). The theorem asserts that for each function ƒ in L2, there is a sequence {mV} 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 

Footnotes

This work was supported by the National Science Foundation through Research Grant NSF G-2789.

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