Abstract
The aim of this paper is to prove the a.e. convergence of sequences of the Cesàro and Riesz means of the Walsh-Fourier series of d variable integrable functions. That is, let a = (a 1, ...,a d ): ℕ → ℕd (d ∈ ℙ) such that a j (n + 1) ≥ δ sup k≤n a j (k) (j = 1, ..., d, n ∈ ℕ) for some δ > 0 and a 1(+∞) = ... = a d (+∞) = +∞. Then, for each integrable function f ∈ L 1(I d), we have the a.e. relation for the Cesàro means limn→∞ σ αa(n) f = f and for the Riesz means limn→∞ σ α,γa(n) f = f for any 0 < α j ≤ 1 ≤ γ j (j = 1, ..., d). A straightforward consequence of our result is the so-called cone restricted a.e. convergence of the multidimensional Cesàro and Riesz means of integrable functions, which was proved earlier by Weisz.
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Supported by project TÁMOP-4.2.2.A-11/1/KONV-2012-0051
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Gát, G. Almost everywhere convergence of sequences of Cesàro and Riesz means of integrable functions with respect to the multidimensional Walsh system. Acta. Math. Sin.-English Ser. 30, 311–322 (2014). https://doi.org/10.1007/s10114-013-1766-3
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DOI: https://doi.org/10.1007/s10114-013-1766-3