dorsal/arxiv
View SchemaGeneralized Hermite Polynomials and the Heat Equation for Dunkl Operators
| Authors | Margit Rösler |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9703006 |
| URL | https://arxiv.org/abs/q-alg/9703006 |
| DOI | 10.1007/s002200050307 |
Abstract
Based on the theory of Dunkl operators, this paper presents a general concept of multivariable Hermite polynomials and Hermite functions which are associated with finite reflection groups on $\b R^N$. The definition and properties of these generalized Hermite systems extend naturally those of their classical counterparts; partial derivatives and the usual exponential kernel are here replaced by Dunkl operators and the generalized exponential kernel K of the Dunkl transform. In case of the symmetric group $S_N$, our setting includes the polynomial eigenfunctions of certain Calogero-Sutherland type operators. The second part of this paper is devoted to the heat equation associated with Dunkl's Laplacian. As in the classical case, the corresponding Cauchy problem is governed by a positive one-parameter semigroup; this is assured by a maximum principle for the generalized Laplacian. The explicit solution to the Cauchy problem involves again the kernel K, which is, on the way, proven to be nonnegative for real arguments.
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"abstract": "Based on the theory of Dunkl operators, this paper presents a general concept\nof multivariable Hermite polynomials and Hermite functions which are associated\nwith finite reflection groups on $\\b R^N$. The definition and properties of\nthese generalized Hermite systems extend naturally those of their classical\ncounterparts; partial derivatives and the usual exponential kernel are here\nreplaced by Dunkl operators and the generalized exponential kernel K of the\nDunkl transform. In case of the symmetric group $S_N$, our setting includes the\npolynomial eigenfunctions of certain Calogero-Sutherland type operators. The\nsecond part of this paper is devoted to the heat equation associated with\nDunkl\u0027s Laplacian. As in the classical case, the corresponding Cauchy problem\nis governed by a positive one-parameter semigroup; this is assured by a maximum\nprinciple for the generalized Laplacian. The explicit solution to the Cauchy\nproblem involves again the kernel K, which is, on the way, proven to be\nnonnegative for real arguments.",
"arxiv_id": "q-alg/9703006",
"authors": [
"Margit R\u00f6sler"
],
"categories": [
"q-alg",
"math.QA"
],
"doi": "10.1007/s002200050307",
"title": "Generalized Hermite Polynomials and the Heat Equation for Dunkl Operators",
"url": "https://arxiv.org/abs/q-alg/9703006"
},
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