The Product Formula and Convolution Structure Associated with the Generalized Hermite Polynomials

✍ Scribed by C. Markett

• Publisher: Elsevier Science
• Year: 1993
• Tongue: English
• Weight: 517 KB
• Volume: 73
• Category: Article
• ISSN: 0021-9045
• DOI: 10.1006/jath.1993.1038

✦ Synopsis

Here the product formula for the generalized and suitably normalized Hermite polynomials with parameter (\mu \geqslant 0) will be explicitly established. Its measure turns out to be absolutely continuous and supported on two disjoint intervals lying symmetrically on the real line, provided that (\mu>0). In the limit case (\mu=0), which is associated with the classical Hermite polynomials, four additional point masses occur at the endpoints of the two intervals. As an application, the product formula is used to introduce a generalized translation operator and a corresponding convolution product on appropriately weighted Lebesgue spaces. To this end, norm estimates of the translation operator from above and below are presented. For any (\mu \geqslant \frac{1}{2}), this gives rise to a quasi-positive convolution algebra. 1993 Academic Press, Inc.