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Summability of multi-dimensional trigonometric Fourier series. (English) Zbl 1285.42010

Summary: 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 \(H_{p}\) to \(L_{p}\), for all \(p > p_{0}\), where \(p_{0}\) 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 \(\theta \)-summability and for Fourier transforms.

MSC:

42B08 Summability in several variables
42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
42B30 \(H^p\)-spaces
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