On the uniform convergence of double sine series

Abstract

The fundamental theorem in the theory of the uniform convergence of sine series is due to Chaundy and Jolliffe from 1916 (see [1]). Several authors gave conditions for this problem supposing that coefficients are monotone, non-negative or more recently, general monotone (see [8], [6] and [2], for example). There are also results for the regular convergence of double sine series to by uniform in case the coefficients are monotone or general monotone double sequences. In this article we give new sufficient conditions for the uniformity of the regular convergence of double sine series, which are necessary as well in case the coefficients are non-negative. We shall generalize those results defining a new class of double sequences for the coefficients.Mathematics subject classification number: 42A20, 42A32, 42B99.

On the Uniform Convergence of Sine Series with Square Root

“…Duzinkiewicz and Szal [7] introduce a new class of double sequence called ( , , , ), which is a generalization of the class considered by Kórus, and they obtain sufficient and necessary conditions for uniform convergence of double sine series.…”

A Note on Derivative of Sine Series with Square Root

“…Duzinkiewicz and Szal [2] introduce a new class of double sequences called DGMðα, β, γ, rÞ, which is a generalization of the class considered by Kórus, and they obtain sufficient and necessary conditions for uniform convergence of double sine series.…”

A necessary condition for uniform convergence of double sine series with $p$-bounded variation coefficients

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We introduce a new class of double sequences called DGM (p, α, β, γ, r), which is a generalization of a class considered by Duzinkiewicz and Szal (2018). We obtain a necessary condition for the uniform convergence of double sine series with coefficients belonging to this class.

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“…He also showed that MVBVDS SBVDS 1 . Next, Duzinkiewicz and Szal [2] defined a new class of sequences: Definition 1.7. A double sequence c := {c jk } ∞ j,k=1 ⊂ C belongs to DGM(α, β, γ, r) (called Double General Monotone), if there exist a positive constant C and an integer λ ≥ 1, depending only on (c jk ) ∞ j,k=1 , for which…”