On Approximation of Derivatives of Functions in Weighted Morrey Space

Abstract

Weighted Morrey space is one of the most important types of Banach spaces. This space was introduced by Y. Komori and S. Shirai in 2009 as the set of all periodic and measurable functions  defined on  for which:

where, ,  and  is a weight function.

Many researches have investigated the properties of Morrey space and the approximation of its functions by trigonometric polynomials such as Fejer sums and de la Vallée poussin sums. In this paper, we estimate the deviation of the fractional derivative of order  for a function  belonging to the weighted Morrey Sobolev space from derivative of the best trigonometric polynomial to  in terms of the best approximation, and in terms of the fractional moduli in the weighted Morrey space. Then, we estimate the norm of the fractional derivative for a function  in the weighted Morrey space. Moreover, an estimation of the difference between the derivative of a function  belonging to the weighted Morrey space and the derivative of the de la Vallée Poussin sums of the function  in terms of the fractional moduli in the weighted Morrey space is obtained.