Necessary and sufficient conditions for mean convergence of orthogonal expansions for Freud weights |
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Authors: | S W Jha D S Lubinsky |
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Institution: | 1. Department of Mathematics, National Central University, 32054, Chung-Li, Taiwan Republic of China 2. Department of Mathematics, University of the Witwatersrand, P.O. Wits 2050, Republic of South Africa
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Abstract: | 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 weightsW
2:=e
–2Q
, by first proving a bound for the difference of the orthonormal polynomials of degreen+1 andn–1 of the weightW
2. 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. |
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Keywords: | AMS classification" target="_blank">AMS classification 41A10 42C15 33E15 |
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