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Herz spaces and restricted summability of Fourier transforms and Fourier series. (English) Zbl 1254.42012

Summary: A general summability method, the so-called \(\theta \)-summability is considered for multi-dimensional Fourier transforms and Fourier series. A new inequality for the Hardy-Littlewood maximal function is verified. It is proved that if the Fourier transform of \(\theta \) is in a Herz space, then the restricted maximal operator of the \(\theta \)-means of a distribution is of weak type (1,1), provided that the supremum in the maximal operator is taken over a cone-like set. From this it follows that \(\sigma _T^{\theta}f \to f\) over a cone-like set a.e. for all \(f \in L_1(\mathbb R^d)\). Moreover, \(\sigma _T^{\theta}f(x)\) converges to \(f(x)\) over a cone-like set at each Lebesgue point of \(f \in L_1(\mathbb R^d\) if and only if the Fourier transform of \(\theta \) is in a suitable Herz space. These theorems are extended to Wiener amalgam spaces as well. The Riesz and Weierstrass summations are investigated as special cases of the \(\theta \)-summation.

MSC:

42B08 Summability in several variables
42B25 Maximal functions, Littlewood-Paley theory
42B35 Function spaces arising in harmonic analysis
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