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Journal of Function Spaces and Applications
Volume 2012 (2012), Article ID 982360, 41 pages
http://dx.doi.org/10.1155/2012/982360
Research Article

Approximating Polynomials for Functions of Weighted Smirnov-Orlicz Spaces

Department of Mathematics, Faculty of Art and Science, Balikesir University, 10145 Balikesir, Turkey

Received 9 July 2009; Accepted 13 December 2009

Academic Editor: V. M. Kokilashvili

Copyright © 2010 Ramazan Akgün. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let 𝐺 0 and 𝐺 ∞ be, respectively, bounded and unbounded components of a plane curve Ξ“ satisfying Dini's smoothness condition. In addition to partial sum of Faber series of 𝑓 belonging to weighted Smirnov-Orlicz space 𝐸 𝑀 , πœ” ( 𝐺 0 ), we prove that interpolating polynomials and Poisson polynomials are near best approximant for 𝑓 . Also considering a weighted fractional moduli of smoothness, we obtain direct and converse theorems of trigonometric polynomial approximation in Orlicz spaces with Muckenhoupt weights. On the bases of these approximation theorems, we prove direct and converse theorems of approximation, respectively, by algebraic polynomials and rational functions in weighted Smirnov-Orlicz spaces 𝐸 𝑀 , πœ” ( 𝐺 0 ) and 𝐸 𝑀 , πœ” ( 𝐺 ∞ ) .