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HERMITE FUNCTION EXPANSIONS VERSUS HERMITE POLYNOMIAL EXPANSIONS

Published online by Cambridge University Press:  23 August 2006

I. ABU-FALAHAH  and
J. L. TORREA
I. ABU-FALAHAH
Affiliation:
Departamento de Matemáticas Facultad de Ciencias Universidad Autónoma de Madrid 28049 Madrid, Spain e-mail: ibraheem.abufalahah@uam.es, joseluis.torrea@uam.es
J. L. TORREA
Affiliation:
Departamento de Matemáticas Facultad de Ciencias Universidad Autónoma de Madrid 28049 Madrid, Spain e-mail: ibraheem.abufalahah@uam.es, joseluis.torrea@uam.es
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Abstract

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We consider expansions with respect to the multi-dimensional Hermite functions and to the multi-dimensional Hermite polynomials. They are respectively eigenfunctions of the Harmonic oscillator $\cal{L}= -\Delta +|x|^2$ and of the Ornstein-Uhlenbeck operator ${\bf L} = -\Delta +2x \cdot \nabla.$ The corresponding heat semigroups and Riesz transforms are considered and results on both aspects (polynomials and functions) are obtained.

Keywords

42C1042B2042B25
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 48 , Issue 2 , May 2006 , pp. 203 - 215
Copyright
2006 Glasgow Mathematical Journal Trust