Necessary and sufficient conditions for mean convergence of orthogonal expansions for Freud weights

A classic 1970 paper of B. Muckenhoupt established necessary and sufficient conditions for weightedL_p convergence of Hermite series, that is, orthogonal expansions corresponding to the Hermite weight. We generalize these to orthogonal expansions for a class of Freud weightsW²:=e^–2Q, by first proving a bound for the difference of the orthonormal polynomials of degreen+1 andn–1 of the weightW². Our identical necessary and sufficient conditions close a slight gap in Muckenhoupt's conditions for the Hermite weight at least forp>1. Moreover, our necessary conditions apply whenQ(x)=|x|, >1 while our sufficient conditions apply at least for =2,4,6,....Communicated by Vilmos Totik.