Convergence of a Subsequence of Triangular Partial Sums of Double Walsh-Fourier Series

G. Gát; U. Goginava

doi:10.3103/S1068362319040034

Convergence of a Subsequence of Triangular Partial Sums of Double Walsh-Fourier Series

G. Gát, U. Goginava

Research output: Contribution to journal › Article › peer-review

1 Citation (Scopus)

Abstract

In 1987 Harris proved-among others that for each 1 ≤ p < 2 there exists a two-dimensional function f ∈ L_p such that its triangular Walsh-Fourier series does not converge almost everywhere. In this paper we prove that the set of the functions from the space L_p(II²) (1 ≤ p < 2) with subsequence of triangular partial means S2AΔ(f) of the double Walsh-Fourier series convergent in measure on II² is of first Baire category in L_p(II²). We also prove that for each function f ∈ L₂(II²) a.e. convergence Sa(n)Δ(f)→f holds, where a(n) is a lacunary sequence of positive integers.

Original language	English
Pages (from-to)	210-215
Number of pages	6
Journal	Journal of Contemporary Mathematical Analysis
Volume	54
Issue number	4
DOIs	https://doi.org/10.3103/S1068362319040034
Publication status	Published - Jul 1 2019
Externally published	Yes

Keywords

42C10
Convergence in measure
Double Walsh-Fourier series
triangular partial sum

ASJC Scopus subject areas

Analysis
Control and Optimization
Applied Mathematics

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10.3103/S1068362319040034

Cite this

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title = "Convergence of a Subsequence of Triangular Partial Sums of Double Walsh-Fourier Series",

abstract = "In 1987 Harris proved-among others that for each 1 ≤ p < 2 there exists a two-dimensional function f ∈ Lp such that its triangular Walsh-Fourier series does not converge almost everywhere. In this paper we prove that the set of the functions from the space Lp(II2) (1 ≤ p < 2) with subsequence of triangular partial means S2AΔ(f) of the double Walsh-Fourier series convergent in measure on II2 is of first Baire category in Lp(II2). We also prove that for each function f ∈ L2(II2) a.e. convergence Sa(n)Δ(f)→f holds, where a(n) is a lacunary sequence of positive integers.",

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AU - Goginava, U.

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N2 - In 1987 Harris proved-among others that for each 1 ≤ p < 2 there exists a two-dimensional function f ∈ Lp such that its triangular Walsh-Fourier series does not converge almost everywhere. In this paper we prove that the set of the functions from the space Lp(II2) (1 ≤ p < 2) with subsequence of triangular partial means S2AΔ(f) of the double Walsh-Fourier series convergent in measure on II2 is of first Baire category in Lp(II2). We also prove that for each function f ∈ L2(II2) a.e. convergence Sa(n)Δ(f)→f holds, where a(n) is a lacunary sequence of positive integers.

AB - In 1987 Harris proved-among others that for each 1 ≤ p < 2 there exists a two-dimensional function f ∈ Lp such that its triangular Walsh-Fourier series does not converge almost everywhere. In this paper we prove that the set of the functions from the space Lp(II2) (1 ≤ p < 2) with subsequence of triangular partial means S2AΔ(f) of the double Walsh-Fourier series convergent in measure on II2 is of first Baire category in Lp(II2). We also prove that for each function f ∈ L2(II2) a.e. convergence Sa(n)Δ(f)→f holds, where a(n) is a lacunary sequence of positive integers.

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KW - Convergence in measure

KW - Double Walsh-Fourier series

KW - triangular partial sum

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Convergence of a Subsequence of Triangular Partial Sums of Double Walsh-Fourier Series

Abstract

Keywords

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