Best approximations and moduli of smoothness of functions and their derivatives in [formula omitted], [formula omitted

Abstract

Several new inequalities for moduli of smoothness and errors of the best approximation of a function and its derivatives in the spaces Lp, 0<p<1, are obtained. For example, it is shown that for any 0<p<1, k,r∈N, and a function f∈W1rωr+k(f,δ)p≤C(p,k,r)δr+1p−1∫0δωk(f(r),t)ppt2−pdt1p.Similar inequalities are obtained for the Ditzian–Totik moduli of smoothness and the error of the best approximation of functions by trigonometric and algebraic polynomials and splines. As an application, positive results about simultaneous approximation of a function and its derivatives by the mentioned approximation methods in the spaces Lp, 0<p<1, are derived.