Necessary and sufficient conditions for mean convergence of orthogonal expansions for Freud weights
In a classic paper, B. Muckenhoupt established necessary and sufficient conditions for weighted \(L_p\) convergence of Hermite series, i.e., orthogonal expansions in Hermite polynomials. In the paper under review, the Hermite polynomials are replaced by systems of orthogonal polynomials that can be considered as natural generalizations. The new polynomials are orthogonal with respect to a class of Freud weights \(W^2:= e^{- 2Q}\). For the convergence of expansions in these polynomials necessary and sufficient conditions are obtained that are identical with those proved by Muckenhoupt. The necessary condition applies when \(Q(x)= |x|^\alpha\), \(\alpha> 1\), while the sufficient one applies for \(\alpha= 2, 4, 6,\dots\;\).
scientific article; zbMATH DE number 810114
scientific article