K-functional, weighted moduli of smoothness, and best weighted polynomial approximation on a simplex

An inverse theorem for the best weighted polynomial approximation of a function inLpw. (S) is established. We also investigate Besov spaces generated by Freud weight and their connection with algebraic polynomial approximation inLp(R)W., wherew . is a Jacobi-type weight onS, 0<p . .,S is a simplex andW . is a Freud weight. For Ditzian-TotikK-functionals onL p(S), 1 .p . ., we obtain a new equivalence expression.