Abstract
A classic 1970 paper of B. Muckenhoupt established necessary and sufficient conditions for weightedLp 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|, α>1 while our sufficient conditions apply at least for α=2,4,6,....
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