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Expansion by orthogonal systems with respect to Freud weights related to Hardy spaces


Author: Z. Ditzian
Journal: Proc. Amer. Math. Soc. 146 (2018), 1665-1672
MSC (2010): Primary 42C10, 42B30, 42C05, 26D15
DOI: https://doi.org/10.1090/proc/13842
Published electronically: December 4, 2017
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Abstract: For the basic class of Freud weights $ w_\alpha (x) = \,\exp (-\vert x\vert ^\alpha /2),\,\alpha >1$ the coefficients of the expansion of $ w_\alpha f\in H_p(R)$ by the Freud orthogonal system $ \{w_\alpha p_{n,\alpha }\}^\infty _{n=0}\,,$ where $ p_{n,\alpha }$ are polynomials of degree $ n,$ are related to the quasi-norm (or norm) of $ w_\alpha f$ in $ H_p(R).$ Relations are achieved for all $ \alpha >1$ and $ \frac 12 <p<1,$ and for some $ \alpha $ for a larger range of $ p.$ As a result, estimates for $ 1<p\le 2$ are also improved.


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Additional Information

Z. Ditzian
Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1
Email: zditzian@gmail.com

DOI: https://doi.org/10.1090/proc/13842
Keywords: Hardy spaces, Freud weights, expansion by orthogonal system, atomic decomposition, Hardy-Littlewood inequality.
Received by editor(s): January 7, 2017
Received by editor(s) in revised form: June 5, 2017
Published electronically: December 4, 2017
Communicated by: Yuan Xu
Article copyright: © Copyright 2017 American Mathematical Society

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