Waterman classes and spherical partial sums of double Fourier series

Dyachenko, Mikhail Ivanovich

doi:10.1007/bf01904022

Cited by 21 publications

(15 citation statements)

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“…It is clear that the functions of ΛBV (A) are bounded, but the question of continuity is more complicated than in the case of functions of one variable. Dyachenko [3] has proved the following theorem.…”

Section: Introductionmentioning

confidence: 97%

Convergence of double Fourier series and $W$-classes

Dyachenko

Waterman

2004

Trans. Amer. Math. Soc.

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Abstract. The double Fourier series of functions of the generalized bounded variation class {n/ ln(n + 1)} * BV are shown to be Pringsheim convergent everywhere. In a certain sense, this result cannot be improved. In general, functions of class Λ * BV, defined here, have quadrant limits at every point and, for f ∈ Λ * BV, there exist at most countable sets P and Q such that, for x / ∈ P and y / ∈ Q, f is continuous at (x, y). It is shown that the previously studied class ΛBV contains essentially discontinuous functions unless the sequence Λ satisfies a strong condition.

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Section: Introductionmentioning

confidence: 97%

Convergence of double Fourier series and $W$-classes

Dyachenko

Waterman

2004

Trans. Amer. Math. Soc.

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“…The convergence of spherical and other partial sums of double Fourier series of functions of bounded ƒ-variation was investigated in detail by Dyachenko (see [4,5] and the references therein).…”

Section: Classes Of Functions Of Bounded Generalized Variationmentioning

confidence: 99%

“…Since then this notion had been generalized by many authors (quadratic variation,ˆ-variation, ƒ-variation, etc., see [1,[3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]). In the two-dimensional case the class BV of functions of bounded variation was introduced by Hardy [11].…”

Section: Classes Of Functions Of Bounded Generalized Variationmentioning

confidence: 99%

Convergence of double Fourier series and generalized Λ-variation

Гогинава¹,

Sahakian²

2012

Georgian Mathematical Journal

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“…Pl.Pljk X i X k Saakyan showed that for / £ HBV(T 2 ), the rectangular partial sums of the However, Dyachenko has shown [6] that in order for the quadrant limits to exist everywhere, it is necessary and sufficient that Σ Aj~2 = 00. Indeed, if Σ λ^2 < oo, then ABV (A) contains an essentially discontinuous function [6].…”

Section: Definition 1 Let F Be a Measurable Function On The Rectanglmentioning

confidence: 99%

On the Magnitude of Fourier Coefficients Ii

Schramm¹,

Waterman²

2004

Analysis

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For functions on [-ιτ,ττ] 2 which are in various generalized variation classes, estimates of the magnitude of the Fourier coefficients are made. The principal result is best possible in a certain sense.The calculation of the order of magnitude of the Fourier coefficients of a function / of bounded variation is usually accomplished by use of integration by parts and the fact that the variation of / is equal to / \df\ [25, vol.1, p.48]. Unfortunately, this method provides no clue as to how to proceed in higher dimensions and for functions of generalized bounded variation.By extending the notion of integration by parts to the double Riemann-Stieltjes integral, Móricz has used this classical method to effect this calculation for functions of Hardy-Krause bounded variation in two variables [12]. Here we will demonstrate a simple method, used by us [17] and by Silei Wang [19] in the one-variable case, which enables us to determine the order of the Fourier coefficients of functions of bounded variation in any finite number of variables and which extends readily to classes of functions of generalized variation, including the W-classes. Using this method, Fülöp and Móricz [11] have since obtained a result which subsumes our first result. They cite a paper of Wu [24], rediscovering this method in the onevariable case.

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Waterman classes and spherical partial sums of double Fourier series

Cited by 21 publications

References 1 publication

Convergence of double Fourier series and $W$-classes

Convergence of double Fourier series and $W$-classes

Convergence of double Fourier series and generalized Λ-variation

On the Magnitude of Fourier Coefficients Ii

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