Approximation of functions in by trigonometric polynomials

Abstract

We consider the Lebesgue space with variable exponent . It consists of measurable functions for which the integral exists. We establish an analogue of Jackson's first theorem in the case when the -periodic variable exponent satisfies the conditionUnder the additional assumption we also get an analogue of Jackson's second theorem. We establish an -analogue of Bernstein's estimate for the derivative of a trigonometric polynomial and use it to prove an inverse theorem for the analogues of the Lipschitz classes for . Thus we establish direct and inverse theorems of the theory of approximation by trigonometric polynomials in the classes . In the definition of the modulus of continuity of a function , we replace the ordinary shift by an averaged shift determined by Steklov's function .