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Cesàro summability of two-dimensional Walsh-Fourier series. (English) Zbl 0857.42016

Using the quasi-local operator approach found in F. Móricz, F. Schipp and the reviewer [Trans. Am. Math. Soc. 329, No. 1, 131-140 (1992; Zbl 0795.42016)], the author shows that the maximal Cesàro operator associated with summability of double Walsh-Fourier series over positive cones is of type \((H^p,L^p)\), for \({1\over 2}<p\leq\infty\), and of weak type \((1,1)\). Among applications is a Walsh analogue of a theorem of Marcinkiewicz and Zygmund: Walsh-Fourier series of \(L^1\) functions are almost everywhere \((C,1)\) summable provided the limit is taken through indices which lie in a positive cone.

MSC:

42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
43A75 Harmonic analysis on specific compact groups
60G42 Martingales with discrete parameter

Citations:

Zbl 0795.42016
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References:

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