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Convergence of the Hausdorff Means ofDouble Fourier Series*

Published online by Cambridge University Press:  20 November 2018

Fred Ustina
Fred Ustina*
Affiliation:
University of Alberta, Edmonton, Alberta
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In this paper we prove that if {sm, n(x, y)} is the sequence of partial sums of the Fourier series of a function f(x, y), which is periodic in each variable and of bounded variation in the sense of Hardy-Krause in the period rectangle, then {sm, n(x, y)} converges uniformly to f(x, y) in any closed region D in which this function is continuous at every point. This result is then used to prove that the regular Hausdorff means of the Fourier series of such a function also converge uniformly in such a region.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 11 , Issue 4 , October 1968 , pp. 585 - 591
Copyright
Copyright © Canadian Mathematical Society 1968

Footnotes

*

The preparation of this paper was financed in part by a Canadian Mathematical Congress summer research grant (1967), and in part by a post-doctoral fellowship at the University of Alberta (1966-67). The author is most grateful to the referee whose earlier comments were basic to the preparation of this paper.

References

1. Hallenbach, F., Zur Theorie der Limitierungsverfahren von Doppelfolgen. Inaugerral-Dissertation, Rheinischen Friedrich-Wilhelms-Universität, Bonn (1933).Google Scholar
2. Hobson, E. W., The theory of functions of a real variable. (Cambridge University Press, 2nd ed., Vol. 2, 1926).Google Scholar
3. McShane, E.J., Integration. (Princeton University Press, 1944).Google Scholar