Uniform convergence of Cesàro means of negative order of double Walsh-Fourier series

Ushangi Goginava

doi:10.1016/S0021-9045(03)00134-5

Uniform convergence of Cesàro means of negative order of double Walsh-Fourier series

Ushangi Goginava

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

22 Citations (Scopus)

Abstract

In this paper we prove that if f ∈ C_W([0, 1]² and the function f is bounded partial p-variation for some p ∈ [1, + ∞) then the double Walsh-Fourier series of the function f is uniformly (C; -α, -β) summable (α + β < 1/p, α, β > 0) in the sense of Pringsheim. If α + β ≥ 1/p then there exists a continuous function f₀ of bounded partial p-variation on [0,1]² such that the Cesàro (C; -α , -β) means σ_n,m^-α,-β (f₀; 0,0) of the double Walsh-Fourier series of f₀ diverge over cubes.

Original language	English
Pages (from-to)	96-108
Number of pages	13
Journal	Journal of Approximation Theory
Volume	124
Issue number	1
DOIs	https://doi.org/10.1016/S0021-9045(03)00134-5
Publication status	Published - Sept 2003
Externally published	Yes

Keywords

Cesàro means
Double Walsh-Fourier series
Uniform summability

ASJC Scopus subject areas

Analysis
Numerical Analysis
General Mathematics
Applied Mathematics

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10.1016/S0021-9045(03)00134-5

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abstract = "In this paper we prove that if f ∈ CW([0, 1]2 and the function f is bounded partial p-variation for some p ∈ [1, + ∞) then the double Walsh-Fourier series of the function f is uniformly (C; -α, -β) summable (α + β < 1/p, α, β > 0) in the sense of Pringsheim. If α + β ≥ 1/p then there exists a continuous function f0 of bounded partial p-variation on [0,1]2 such that the Ces{\`a}ro (C; -α , -β) means σn,m-α,-β (f0; 0,0) of the double Walsh-Fourier series of f0 diverge over cubes.",

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AB - In this paper we prove that if f ∈ CW([0, 1]2 and the function f is bounded partial p-variation for some p ∈ [1, + ∞) then the double Walsh-Fourier series of the function f is uniformly (C; -α, -β) summable (α + β < 1/p, α, β > 0) in the sense of Pringsheim. If α + β ≥ 1/p then there exists a continuous function f0 of bounded partial p-variation on [0,1]2 such that the Cesàro (C; -α , -β) means σn,m-α,-β (f0; 0,0) of the double Walsh-Fourier series of f0 diverge over cubes.

KW - Cesàro means

KW - Double Walsh-Fourier series

KW - Uniform summability

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Uniform convergence of Cesàro means of negative order of double Walsh-Fourier series

Abstract

Keywords

ASJC Scopus subject areas

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