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Triangular summability of two-dimensional Fourier transforms
Authors:Ferenc Weisz
Affiliation:1. Department of Numerical Analysis, E?tv?s L. University, H-1117, Budapest, P??zm??ny P. s??t??ny 1/C., Hungary
Abstract:We consider the triangular summability of two-dimensional Fourier transforms, and show that the maximal operator of the triangular-??-means of a tempered distribution is bounded from H p (?2) to L p (?2) for all 2/(2 + ??) < p ?? ??; consequently, it is of weak type (1,1), where 0 < ?? ?? 1 is depending only on ??. As a consequence, we obtain that the triangular-??-means of a function f ?? L 1(?2) converge to f a.e. Norm convergence is also considered, and similar results are shown for the conjugate functions. Some special cases of the triangular-??-summation are considered, such as the Weierstrass, Picar, Bessel, Fejér, de la Vallée-Poussin, Rogosinski, and Riesz summations.
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