Abstract
Under some conditions on \({\theta}\), we characterize the set of convergence of the Marcinkiewicz-\({\theta\mbox{-}}\)means of a function \({f \in L_1(\mathbb{T}^d)}\). More exactly, the \({\theta\mbox{-}}\)means converge to f at each modified strong Lebesgue point. The same holds for a weaker version of Lebesgue points, for the so called modified Lebesgue points of \({f \in L_p(\mathbb{T}^d)}\), whenever \({1 < p < \infty}\). The \({\theta\mbox{-}}\)summability includes the Fejér, Abel, Cesàro and some other summations. As an application we give simple proofs for the classical one-dimensional strong summability results of Hardy and Littlewood, Marcinkiewicz, Zygmund and Gabisoniya and generalize them for strong \({\theta\mbox{-}}\)summability.
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This research was supported by the Hungarian Scientific Research Funds (OTKA) No. K115804.
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Weisz, F. Marcinkiewicz summability of Fourier series, Lebesgue points and strong summability. Acta Math. Hungar. 153, 356–381 (2017). https://doi.org/10.1007/s10474-017-0737-z
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DOI: https://doi.org/10.1007/s10474-017-0737-z
Key words and phrases
- multi-dimensional Fourier series
- Fejér summability
- \({\theta\mbox{-}}\)summability
- Marcinkiewicz summability
- Lebesgue point
- strong summability