On the Rate of Convergence of Singular Integrals for Hölder Continuous Functions

“…Therefore the consideration of approximation problems in various norms is useful.Recently in many papers approximation problems have been studied in the Holder normsIn [2] the orders of approximation are obtained for the Picard, Poisson-Cauchy and GauD-WeierstraD singular integrals in the supremum norm and the classical Holder norm. In [3] error estimates for the Fourier sum and the trigonometric interpolatory polynomial are given in generalized Holder norms.Using notations given in [2] and [3], we present in this paper approximation theorems for the singular integrals [2] in some generalized Holder norms. Those theorems contain and improve the results obtained in [2].…”

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confidence: 99%

On Some Singular Integrals in Hölder Spaces

Firlejy

Rempulska

1994

Mathematische Nachrichten

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The approximation of functions by singular integrals is an important question in the theory of differential and integral equations. Therefore the consideration of approximation problems in various norms is useful.Recently in many papers approximation problems have been studied in the Holder norms In [2] the orders of approximation are obtained for the Picard, Poisson-Cauchy and GauD-WeierstraD singular integrals in the supremum norm and the classical Holder norm. In [3] error estimates for the Fourier sum and the trigonometric interpolatory polynomial are given in generalized Holder norms. Using notations given in [2] and [3], we present in this paper approximation theorems for the singular integrals [2] in some generalized Holder norms. Those theorems contain and improve the results obtained in [2]. ([I -41). 1. Notations 1.1. Let C be the space of real-valued functions, uniformly continuous and bounded on the real line R and let (1) llfllc = SUP If(4l. xeR For given f E C let m(.; f) be the modulus of continuity, i.e., m ( t ; f ) = S U P ll4fllc, t 2 0 ,

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“…The rate of convergence of singular integrals has been studied earlier in [5], [12], [13], [15], [3], [6], [7], [8], [9] and these articles motivate this work. Here we consider some very general singular integral operators over R and we study the degree of approximation to the unit operator with rates over smooth functions.…”

Section: Introductionmentioning

confidence: 99%

Uniform Convergence With Rates of General Singular Operators

Anastassiou

Mezei

2013

Cubo

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In this article we study the approximation properties of general singular integral operators over the real line. We establish their convergence to the unit operator with rates. The estimates are mostly sharp and they are pointwise or uniform. The established inequalities involve the higher order modulus of smoothness. We apply this theory to the trigonometric singular operators. RESUMENEn este artículo estudiamos propiedades de aproximación de operadores integrales singulares generales sobre la recta real. Establecemos su convergencia al operador unidad con las tasas correspondientes. Las estimaciones son mayormente ajustadas y son tanto puntuales como uniformes. Las desigualdades encontradas involucran el módulo de suavidad de alto orden. Aplicamos esta teoría a los operadores singulares trigonométricos.

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On the Rate of Convergence of Singular Integrals for Hölder Continuous Functions

Cited by 31 publications

References 6 publications

On modified Picard and Gauss—Weierstrass singular integrals

On modified Picard and Gauss—Weierstrass singular integrals

On Some Singular Integrals in Hölder Spaces

Uniform Convergence With Rates of General Singular Operators

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