Resumen de Direct theorems of trigonometric approximation for variable exponent Lebesgue spaces

Ramazan Akgün

• Jackson type direct theorems are considered in variable exponent Lebesgue spaces Lp(x) with exponent p(x) satisfying 1≤esinfx∈0,2π]p(x), esssup∈[0,2π]p(x)<∞, 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 Zvf(⋅):=1v∫v0f(⋅+t)dt in these spaces. We give the main properties of the modulus of smoothness Ωr(f,v)p(⋅):=∥(I−Zv)rf∥p(⋅)(∈N)in Lp(x), where I is the identity operator. An equivalence of the modulus of smoothness and Peetre's K-functional is established.