Taiwanese Journal of Mathematics

APPROXIMATION PROPERTIES OF POISSON INTEGRALS FOR ORTHOGONAL EXPANSIONS

Mehmet Ali O¨ zarslan and Oktay Duman

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Abstract

In the present paper we introduce Poisson type integrals for orthogonal expansions. We first give some direct computations for the moments and compute the rates of convergence by means of the modulus of continuity and the Lipschitz functionals; and also we prove that our results are stronger and more general than the results obtained by Toczek and Wachnicki [J. Approx. Theory 116 (2002), 113-125]. We obtain a statistical approximation theorem by using the concept of T−statistical convergence which is a (non-matrix) summability transformation. Furthermore, we give a general Voronovskaya type theorem for these operators. Finally, introducing a higher order generalization of Poisson integrals we discuss their approximation properties.

Article information

Source
Taiwanese J. Math., Volume 12, Number 5 (2008), 1147-1163.

Dates
First available in Project Euclid: 20 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500574254

Digital Object Identifier
doi:10.11650/twjm/1500574254

Mathematical Reviews number (MathSciNet)
MR2431886

Subjects
Primary: 41A25: Rate of convergence, degree of approximation 41A36: Approximation by positive operators

Keywords
Poisson integrals orthogonal polynomials positive linear operators $L^{p}$ space modulus of continuity Lipschitz class functionals the Voronovskaya theorem $T-$statistical convergence

Citation

O¨ zarslan, Mehmet Ali; Duman, Oktay. APPROXIMATION PROPERTIES OF POISSON INTEGRALS FOR ORTHOGONAL EXPANSIONS. Taiwanese J. Math. 12 (2008), no. 5, 1147--1163. doi:10.11650/twjm/1500574254. https://projecteuclid.org/euclid.twjm/1500574254


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