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Эта публикация цитируется в 3 научных статьях (всего в 3 статьях)
Vector-Valued Polynomials and a Matrix Weight Function with $B_2$-Action
Charles F. Dunkl Department of Mathematics, University of Virginia,
PO Box 400137, Charlottesville VA 22904-4137, USA
Аннотация:
The structure of orthogonal polynomials on $\mathbb{R}^{2}$ with the weight function $\vert x_{1}^{2}-x_{2}^{2}\vert ^{2k_{0}}\vert x_{1}x_{2}\vert ^{2k_{1}}e^{-( x_{1}^{2}+x_{2}^{2}) /2}$ is based on the Dunkl operators of type $B_{2}$. This refers to the full symmetry group of the square, generated by reflections in the lines $x_{1}=0$ and $x_{1}-x_{2}=0$. The weight function is integrable if $k_{0},k_{1},k_{0} +k_{1}>-\frac{1}{2}$. Dunkl operators can be defined for polynomials taking values in a module of the associated reflection group, that is, a vector space on which the group has an irreducible representation. The unique $2$-dimensional representation of the group $B_{2}$ is used here. The specific operators for this group and an analysis of the inner products on the harmonic vector-valued polynomials are presented in this paper. An orthogonal basis for the harmonic polynomials is constructed, and is used to define an exponential-type kernel. In contrast to the ordinary scalar case the inner product structure is positive only when $( k_{0},k_{1}) $ satisfy $-\frac{1}{2}<k_{0}\pm k_{1}<\frac{1}{2}$. For vector polynomials $( f_{i}) _{i=1}^{2}$, $( g_{i}) _{i=1}^{2}$ the inner product has the form $\iint_{\mathbb{R}^{2}}f(x) K(x) g(x) ^{T}e^{-( x_{1}^{2}+x_{2}^{2}) /2}dx_{1}dx_{2}$ where the matrix function $K(x)$ has to satisfy various transformation and boundary conditions. The matrix $K$ is expressed in terms of hypergeometric functions.
Ключевые слова:
matrix Gaussian weight function; harmonic polynomials.
Поступила: 16 октября 2012 г.; в окончательном варианте 23 января 2013 г.; опубликована 30 января 2013 г.
Образец цитирования:
Charles F. Dunkl, “Vector-Valued Polynomials and a Matrix Weight Function with $B_2$-Action”, SIGMA, 9 (2013), 007, 23 pp.
Образцы ссылок на эту страницу:
https://www.mathnet.ru/rus/sigma790 https://www.mathnet.ru/rus/sigma/v9/p7
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