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Hardy spaces and Cesàro means of two-dimensional Fourier series. (English) Zbl 0866.42019

Vértesi, P. (ed.) et al., Approximation theory and function series. Proceedings of the international conference dedicated to Károly Tandori on his 70th birthday, Budapest, Hungary, August 21–25, 1995. Budapest: János Bolyai Mathematical Society. Bolyai Soc. Math. Stud. 5, 353-367 (1996).
This article contains a summary of the author’s recent results on two-dimensional Cesàro summability of trigonometric and Walsh-Fourier series together with an indication of how to generalize these results to Vilenkin systems of bounded type. The author includes a sufficiently efficient decomposition of the Fejér kernel (in terms of translates of Vilenkin-Dirichlet kernels) and precise statements of the results. In particular, if \(f\) is integrable on the unit square and \(\sigma_{n,m}f\) represents the Cesàro means of the double Vilenkin-Fourier series of \(f\) (where the underlying Vilenkin system is of bounded type), then \(\sigma_{n,m}f\to f\) a.e. as \(\min(n,m)\to\infty\) provided the pair \((n,m)\) remains in some positive cone (i.e., bounded away from the positive axes).
There are several delicate details which need to be adjusted before the earlier techniques can be used in this sitution. Although full details are not given, the author does a careful job of identifying the necessary adjustments.
For the entire collection see [Zbl 0855.00022].

MSC:

42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
42B30 \(H^p\)-spaces
42B25 Maximal functions, Littlewood-Paley theory
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