Direct theorems of trigonometric approximation for variable exponent Lebesgue spaces

Abstract

Jackson type direct theorems are considered in variable exponent Lebesgue spaces L-p(x) with exponent p(x) satisfying 1 <= ess inf(x is an element of[0,2 pi]) p(x), ess sup(x is an element of[0,2 pi]) p(x) < infinity, and the Dini-Lipschitz condition. Jackson type direct inequalities of trigonometric approximation are obtained for the modulus of smoothness based on one sided Steklov averages

Z(v)f(.) : = 1/v integral(v)(0) f(. + t)dt

in these spaces. We give the main properties of the modulus of smoothness

Omega(r)(f, v)(p(.)) : = parallel to(I - Z(v))(r) f parallel to(p(.)) (r is an element of N)

in L-p(x), where I is the identity operator. An equivalence of the modulus of smoothness and Peetre's K-functional is established.