Smoothness and Best Approximation with Respect to Freud-Type Weights, a New Approach

Abstract

A new measure of smoothness is defined and related to best approximation by polynomials with respect to weighted L p (R) with Freud-type weights. Other related norms are also discussed. Comparisons with the known measure of smoothness on weighted L p spaces are obtained. Related K-functionals and realization functionals are introduced. The new measure of smoothness allows us to consider a more general class of function spaces, to achieve Marchaud, Jackson and Bernstein-type inequalities, and to relate it to expressions involving the coefficients of the expansion by orthogonal polynomials with respect to Freud-type weights. Some of the results are new for approximation by Hermite polynomials in the weighted L p space with the weight $${e^{-x^{2}}}$$ .