Surveys in Approximation Theory, 7 (2012), 1-179.
Summability of Multi-Dimensional Trigonometric Fourier Series
Abstract. We consider the summability of one- and multi-dimensional trigonometric Fourier series. The Fejér and Riesz summability methods are investigated in detail. Different types of summation and convergence are considered. We will prove that the maximal operator of the summability means is bounded from the Hardy space Hp to Lp, for all p > p0, where p0 depends on the summability method and the dimension. For p = 1 we obtain a weak type inequality by interpolation, which ensures the almost everywhere convergence of the summability means. Similar results are formulated for the more general θ-summability and for Fourier transforms.
Published: 1 June 2012.
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Department of Numerical Analysis
Eötvös L. University
H-1117 Budapest, Pázmány P. sétány 1/C.