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= (a1, . . . ,ad) : N → Nd (i>d ∈ P) such that aj(n + 1) ≥ δ supk≤n aj(k) (j = 1, . . . , d, n ∈ N) for some δ > 0 and a1(+∞) = . . . = ad(+∞) = +∞. Then, for each integrable function f ∈ L1(Id), we have the a.e. relation for the Cesàro means limn→∞ σa(n)αf = f and for the Riesz means limn→∞ σa(n)α,γf = f for any 0 < αj ≤ 1 ≤ γj (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.
Key words
Walsh system /
d-dimensional Fejér and Riesz means /
subsequence /
almost everywhere convergence
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References
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Footnotes
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Funding
Supported by project TÁMOP-4.2.2.A-11/1/KONV-2012-0051
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