Preface to the second edition Preface to the first edition 1. Background 2. Orthogonal polynomials in two variables 3. General properties of orthogonal polynomials in several variables 4. Orthogonal polynomials on the unit sphere 5. Examples of orthogonal polynomials in several variables 6. Root systems and Coxeter groups 7. Spherical harmonics associated with reflection groups 8. Generalized classical orthogonal polynomials 9. Summability of orthogonal expansions 10. Orthogonal polynomials associated with symmetric groups 11. Orthogonal polynomials associated with octahedral groups and applications References Author index Symbol index Subject index.

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Orthogonal Polynomials of Several Variables

THE criticism on the passage quoted from p. 3 of the book by Profs. Harkness and Morley (NATURE, February 23, p. 347) turns on the fact that, in dealing with number divorced from measurement, the authors have used the phrase “an infinity of objects” without an explicit statement of its meaning. I am not sure that I understand the passage in their letter which refers to this point; but it seems to me to imply that the distinction between “finite” and “infinite” is one which does not require definition. This is not the only accepted view. It is not, for instance, the view taken in Herr Dedekind's book, “Was sind und was sollen die Zahlen.” As regards the opening sentences of Chapter xv., the authors have apparently misunderstood the point of my objection. With the usually received definition of convergence of an infinite product, Π(1-αn), if convergent, is different from zero. So far as the passage quoted goes, Π(1-αn) might be zero; and it is therefore not shown to be convergent, if the usual definition of convergence be assumed. As to the passage quoted from p. 232, I must express to the authors my regret for having overlooked the fact that the particular rearrangement, there made use of, has been fully justified in Chapter viii. Whether Log x is or is not, at the beginning of Chapter iv., defined by means of a string and a cone, will be obvious to any one who will read the whole passage (p. 46, line 16, to p. 47, line 9) leading up to the definition.

Theory of Functions

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

Dunkl Operators: Theory and Applications

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 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 the 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.

Generalized hermite polynomials and the heat equation for dunkl operators

GROUPS AND GEOMETRIC ANALYSIS Integral Geometry, Invariant Differential Operators, and Spherical Functions (Pure and Applied Mathematics: A Series of Monographs and Textbooks)

For a finite reflection group on $\b R^N,$ the associated Dunkl operators are parametrized first-order differential-difference operators which generalize the usual partial derivatives They generate a commutative algebra which is - under weak assumptions - intertwined with the algebra of partial differential operators by a unique linear and homogeneous isomorphism on polynomials In this paper it is shown that for non-negative parameter values, this intertwining operator is positivity-preserving on polynomials and allows a positive integral representation on certain algebras of analytic functions This result in particular implies that the generalized exponential kernel of the Dunkl transform is positive-definite

Positivity of dunkl's intertwining operator

Markov Processes Related with Dunkl Operators

It is an open conjecture that generalized Bessel functions associated with root systems have a positive product formula for nonnegative multiplicity parameters of the associated Dunkl operators. In this paper, a partial result towards this conjecture is proven, namely a positive radial product formula for the non-symmetric counterpart of the generalized Bessel function, the Dunkl kernel. Radial here means that one of the factors in the product formula is replaced by its mean over a sphere. The key to this product formula is a positivity result for the Dunkl-type spherical mean operator. It can also be interpreted in the sense that the Dunkl-type generalized translation of radial functions is positivity-preserving. As an application, we construct Dunkl-type homogeneous Markov processes associated with radial probability distributions.

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Margit Rösler

Papers

Dunkl Operators: Theory and Applications

Generalized hermite polynomials and the heat equation for dunkl operators

Positivity of dunkl's intertwining operator

Markov Processes Related with Dunkl Operators

A positive radial product formula for the Dunkl kernel