Sharp Jackson and converse theorems of trigonometric approximation in weighted Lebesgue spaces

Abstract

In the present work we prove that improved Jackson type direct theorem of trigonometric polynomial approximation in Lebesgue spaces with Muckenhoupt weights with respect to fractional order moduli of smoothness holds. In addition, we obtain sharp converse and Marchaud inequalities of trigonometric approximation of functions and its fractional derivatives in these weighted Lebesgue spaces.