Approximating functions in the power-type weighted variable exponent Sobolev space by the hardy averaging operator

We investigate the problem of approximating function f in the power-type weighted variable exponent Sobolev space Wr,p(.) ?(.) (0,1), (r = 1, 2, ...), by the Hardy averaging operator A (f) (x) = 1/x ?x0 f(t)dt. If the function f lies in the power-type weighted variable exponent Sobolev space Wr,p(.) ?(.)(0, 1), it is shown that A||(f)?f|| p(.),?(.)?rp(.) ? C ||f(r) p(.),?(.) , where C is a positive constant. Moreover, we consider the problem of boundedness of Hardy averaging operator A in power-type weighted variable exponent grand Lebesgue spaces Lp(.),? ?(.)(0,1). The sufficient criterion established on the power-type weight function ?(.) and exponent p(.) for the Hardy averaging operator to be bounded in these spaces.