Approximating Polynomials for Functions of Weighted Smirnov-Orlicz Spaces

Abstract

Let 𝐺0 and 𝐺∞ be, respectively, bounded and unbounded components of a plane curve Γ satisfying Dini's smoothness condition. In addition to partial sum of Faber series of 𝑓 belonging to weighted Smirnov-Orlicz space 𝐸𝑀,𝜔 (𝐺0), 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 𝐸𝑀,𝜔(𝐺0) and 𝐸𝑀,𝜔(𝐺∞).