Dunkl Operators: Theory and Applications

By M. Rösler. 2002

365 citations recorded by the source corpus.

These lecture notes are intended as an introduction to the theory of rational Dunkl operators and the associated special functions, with an emphasis on positivity and asymptotics. We start with an outline of the general concepts: Dunkl operators, the intertwining operator, the Dunkl kernel and the Dunkl transform. We point out the connection with integrable particle systems of Calogero-Moser-Sutherland type, and discuss some systems of orthogonal polynomials associated with them. A major part is devoted to positivity results for the intertwining operator and the Dunkl kernel, the Dunkl-type heat semigroup, and related probabilistic aspects. The notes conclude with recent results on the asymptotics of the Dunkl kernel.

Subfields: Applied Mathematics, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/5f492d66b44fd0ee6ad5cb1ce1af2f7c48aa56a3

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In Applied Mathematics

In Discrete Mathematics and Combinatorics

Citations in the corpus

Equivalence of K-functionals and modulus of smoothness constructed by generalized Dunkl translations (2008, background): “Here we consider only the Dunkl operators, whose rank equals 1 (i.e., on R 1 ), see [ 11 ] for the general case.”
Hörmander's multiplier theorem for the Dunkl transform (2018, background): “For details we refer the reader to [11], [21], and [22].”
Positive $$L^p$$Lp-Bounded Dunkl-Type Generalized Translation Operator and Its Applications (2017, background): “de with Section 9, where we prove the inverse theorems in Lp-spaces with the Dunkl weight. 2. Notation In this section, we recall the basic notation and results of Dunkl harmonic analysis, see, e.g., [44]. Throughout the paper, hx,yi denotes the standard Euclidean scalar product in d-dimensional Euc”