We consider the generators L k of Heckman-Opdam diffusion processes in the compact and non-compac... more We consider the generators L k of Heckman-Opdam diffusion processes in the compact and non-compact case in N dimensions for root systems of type A and B, with a multiplicity function of the form k = κk 0 with some fixed value k 0 and a varying constant κ ∈ [0, ∞[. Using elementary symmetric functions, we present polynomials which are simultaneous eigenfunctions of the L k for all κ ∈ ]0, ∞[. This leads to martingales associated with the Heckman-Opdam diffusions (X t,1 ,. .. , X t,N) t≥0. As our results extend to the freezing case κ = ∞ with a deterministic limit after some renormalization, we find formulas for the expectations E(N j=1 (y − X t,j)), y ∈ C.
Let V k denote Dunkl's intertwining operator for the root sytem Bn with multiplicity k = (k 1 , k... more Let V k denote Dunkl's intertwining operator for the root sytem Bn with multiplicity k = (k 1 , k 2) with k 1 ≥ 0, k 2 > 0. It was recently shown that the positivity of the operator V k ′ ,k = V k ′ • V −1 k which intertwines the Dunkl operators associated with k and k ′ = (k 1 + h, k 2) implies that h ∈ [k 2 (n−1), ∞[ ∪ ({0, k 2 ,. .. , k 2 (n−1)}−Z +). This is also a necessary condition for the existence of positive Sonine formulas between the associated Bessel functions. In this paper we present two partial converse positive results: For k 1 ≥ 0, k 2 ∈ {1/2, 1, 2} and h > k 2 (n − 1), the operator V k ′ ,k is positive when restricted to functions which are invariant under the Weyl group, and there is an associated positive Sonine formula for the Bessel functions of type Bn. Moreover, the same positivity results hold for arbitrary k 1 ≥ 0, k 2 > 0 and h ∈ k 2 • Z +. The proof is based on a formula of Baker and Forrester on connection coefficients between multivariate Laguerre polynomials and an approximation of Bessel functions by Laguerre polynomials.
For a finite reflection group on R N , the associated Dunkl operators are parametrized firstorder... more For a finite reflection group on R N , the associated Dunkl operators are parametrized firstorder 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.
Transactions of the American Mathematical Society, Jun 17, 2015
The Heckman-Opdam hypergeometric functions of type BC extend classical Jacobi functions in one va... more The Heckman-Opdam hypergeometric functions of type BC extend classical Jacobi functions in one variable and include the spherical functions of non-compact Grassmann manifolds over the real, complex or quaternionic numbers. There are various limit transitions known for such hypergeometric functions. In the present paper, we use an explicit form of the Harish-Chandra integral representation as well as an interpolated variant in order to obtain two limit results, each of them for three continuous classes of hypergeometric functions of type BC which extend the group cases over the fields R, C, H. These limits are distinguished from the known results by explicit and uniform error bounds. The first limit realizes the approximation of the spherical functions of infinite dimensional Grassmannians of fixed rank; here hypergeometric functions of type A appear as limits. The second limit is a contraction limit towards Bessel functions of Dunkl type.
There exist several multivariate extensions of the classical Sonine integral representation for B... more There exist several multivariate extensions of the classical Sonine integral representation for Bessel functions of some index µ + ν with respect to such functions of lower index µ. For Bessel functions on matrix cones, Sonine formulas involve beta densities βµ,ν on the cone and trace already back to Herz. The Sonine representations known so far on symmetric cones are restricted to continuous ranges ℜµ, ℜν > µ 0 , where the involved beta densities are probability measures and the limiting index µ 0 ≥ 0 depends on the rank of the cone. It is zero only in the one-dimensional case, but larger than zero in all multivariate cases. In this paper, we study the extension of Sonine formulas for Bessel functions on symmetric cones to values of ν below the critical limit µ 0. This is achieved by an analytic extension of the involved beta measures as tempered distributions. Following recent ideas by A. Sokal for Riesz distributions on symmetric cones, we analyze for which indices the obtained beta distributions are still measures. At the same time, we characterize the indices for which a Sonine formula between the related Bessel functions exists. As for Riesz distributions, there occur gaps in the admissible range of indices which are determined by the so-called Wallach set.
Transactions of the American Mathematical Society, Jan 14, 2003
It is an open conjecture that generalized Bessel functions associated with root systems have a po... more 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.
The Heckman-Opdam hypergeometric functions of type BC extend classical Jacobi functions in one va... more The Heckman-Opdam hypergeometric functions of type BC extend classical Jacobi functions in one variable and include the spherical functions of non-compact Grassmann manifolds over the real, complex or quaternionic numbers. There are various limit transitions known for such hypergeometric functions, see e.g. [dJ], [RKV]. In the present paper, we use an explicit form of the Harish-Chandra integral representation as well as an interpolated variant, in order to obtain limit results for three continuous classes of hypergeometric functions of type BC which are distinguished by explicit, sharp and uniform error bounds. The first limit realizes the approximation of the spherical functions of infinite dimensional Grassmannians of fixed rank; here hypergeometric functions of type A appear as limits. The second limit is a contraction limit towards Bessel functions of Dunkl type.
In this paper we introduce probability-preserving convolution algebras on cones of positive semid... more In this paper we introduce probability-preserving convolution algebras on cones of positive semidefinite matrices over one of the division algebras F = R, C or H which interpolate the convolution algebras of radial bounded Borel measures on a matrix space M p,q (F) with p q. Radiality in this context means invariance under the action of the unitary group U p (F) from the left. We obtain a continuous series of commutative hypergroups whose characters are given by Bessel functions of matrix argument. Our results generalize well-known structures in the rank one case, namely the Bessel-Kingman hypergroups on the positive real line, to a higher rank setting. In a second part of the paper, we study structures depending only on the matrix spectra. Under the mapping r → spec(r), the convolutions on the underlying matrix cone induce a continuous series of hypergroup convolutions on a Weyl chamber of type B q. The characters are now Dunkl-type Bessel functions. These convolution algebras on the Weyl chamber naturally extend the harmonic analysis for Cartan motion groups associated with the Grassmann manifolds U (p, q)/(U p × U q) over F.
Dunkl operators are differential-difference operators on R N which generalize partial derivatives... more Dunkl operators are differential-difference operators on R N which generalize partial derivatives. They lead to generalizations of Laplace operators, Fourier transforms, heat semigroups, Hermite polynomials, and so on. In this paper we introduce two systems of biorthogonal polynomials with respect to Dunkl's Gaussian distributions in a quite canonical way. These systems, called Appell systems, admit many properties known from classical Hermite polynomials, and turn out to be useful for the analysis of Dunkl's Gaussian distributions. In particular, these polynomials lead to a new proof of a generalized formula of Macdonald due to Dunkl. The ideas for this paper are taken from recent works on non-Gaussian white noise analysis and from the umbral calculus.
It was recently shown by the authors that deformations of hypergroup convolutions w.r.t. positive... more It was recently shown by the authors that deformations of hypergroup convolutions w.r.t. positive semicharacters can be used to explain probabilistic connections between the Gelfand pairs (SL(d, C), SU (d)) and Hermitian matrices. We here study connections between general convolution semigroups on commutative hypergroups and their deformations. We are able to develop a satisfying theory, if the underlying positive semicharacter has some growth property. We present several examples which indicate that this growth condition holds in many interesting cases.
If G is a closed subgroup of a commutative hypergroup K, then the coset space K/G carries a quoti... more If G is a closed subgroup of a commutative hypergroup K, then the coset space K/G carries a quotient hypergroup structure. In this paper, we study related convolution structures on K/G coming from deformations of the quotient hypergroup structure by certain functions on K which we call partial characters with respect to G. They are usually not probability-preserving, but lead to so-called signed hypergroups on K/G. A first example is provided by the Laguerre convolution on [0, ∞[, which is interpreted as a signed quotient hypergroup convolution derived from the Heisenberg group. Moreover, signed hypergroups associated with the Gelfand pair U (n, 1), U (n) are discussed.
We study convolution algebras associated with Heckman-Opdam polynomials. For root systems of type... more We study convolution algebras associated with Heckman-Opdam polynomials. For root systems of type BC we derive three continuous classes of positive convolution algebras (hypergroups) by interpolating the double coset convolution structures of compact Grassmannians U/K with fixed rank over the real, complex or quaternionic numbers. These convolution algebras are linked to explicit positive product formulas for Heckman-Opdam polynomials of type BC, which occur for certain discrete multiplicities as the spherical functions of U/K. The results complement those of [21] for the non-compact case.
Proceedings of the American Mathematical Society, 1999
There exists a generalized Hankel transform of order α ≥ −1/2 on R, which is based on the eigenfu... more There exists a generalized Hankel transform of order α ≥ −1/2 on R, which is based on the eigenfunctions of the Dunkl operator Tαf (x) = f (x) + α + 1 2 f(x) − f(−x) x , f ∈ C 1 (R). For α = −1/2 this transform coincides with the usual Fourier transform on R. In this paper the operator Tα replaces the usual first derivative in order to obtain a sharp uncertainty principle for generalized Hankel transforms on R. It generalizes the classical Weyl-Heisenberg uncertainty principle for the position and momentum operators on L 2 (R); moreover, it implies a Weyl-Heisenberg inequality for the classical Hankel transform of arbitrary order α ≥ −1/2 on [0, ∞[.
We present an explicit product formula for the spherical functions of the compact Gelfand pairs (... more We present an explicit product formula for the spherical functions of the compact Gelfand pairs (G, K 1) = (SU (p + q), SU (p) × SU (q)) with p ≥ 2q, which can be considered as the elementary spherical functions of one-dimensional K-type for the Hermitian symmetric spaces G/K with K = S(U (p) × U (q)). Due to results of Heckman, they can be expressed in terms of Heckman-Opdam Jacobi polynomials of type BC q with specific half-integer multiplicities. By analytic continuation with respect to the multiplicity parameters we obtain positive product formulas for the extensions of these spherical functions as well as associated compact and commutative hypergroup structures parametrized by real p ∈]2q − 1, ∞[. We also obtain explicit product formulas for the involved continuous two-parameter family of Heckman-Opdam Jacobi polynomials with regular, but not necessarily positive multiplicities. The results of this paper extend well known results for the disk convolutions for q = 1 to higher rank.
Stochastic Processes and their Applications, Aug 1, 1991
Optimal or asymptotically optimal linear unbiased mean estimators for a wide class of weakly stat... more Optimal or asymptotically optimal linear unbiased mean estimators for a wide class of weakly stationary processes including ARMA processes are derived explicitly. Furthermore their convergence rates are given.
In this paper, we derive explicit product formulas and positive convolution structures for three ... more In this paper, we derive explicit product formulas and positive convolution structures for three continuous classes of Heckman-Opdam hypergeometric functions of type BC. For specific discrete series of multiplicities these hypergeometric functions occur as the spherical functions of non-compact Grassmann manifolds G/K over one of the skew fields F = R, C, H. We write the product formula of these spherical functions in an explicit form which allows analytic continuation with respect to the parameters. In each of the three cases, we obtain a series of hypergroup algebras which include the commutative convolution algebras of K-biinvariant functions on G as special cases. The characters are given by the associated hypergeometric functions.
These lecture notes are intended as an introduction to the theory of rational Dunkl operators and... more 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.
We consider the generators L k of Heckman-Opdam diffusion processes in the compact and non-compac... more We consider the generators L k of Heckman-Opdam diffusion processes in the compact and non-compact case in N dimensions for root systems of type A and B, with a multiplicity function of the form k = κk 0 with some fixed value k 0 and a varying constant κ ∈ [0, ∞[. Using elementary symmetric functions, we present polynomials which are simultaneous eigenfunctions of the L k for all κ ∈ ]0, ∞[. This leads to martingales associated with the Heckman-Opdam diffusions (X t,1 ,. .. , X t,N) t≥0. As our results extend to the freezing case κ = ∞ with a deterministic limit after some renormalization, we find formulas for the expectations E(N j=1 (y − X t,j)), y ∈ C.
Let V k denote Dunkl's intertwining operator for the root sytem Bn with multiplicity k = (k 1 , k... more Let V k denote Dunkl's intertwining operator for the root sytem Bn with multiplicity k = (k 1 , k 2) with k 1 ≥ 0, k 2 > 0. It was recently shown that the positivity of the operator V k ′ ,k = V k ′ • V −1 k which intertwines the Dunkl operators associated with k and k ′ = (k 1 + h, k 2) implies that h ∈ [k 2 (n−1), ∞[ ∪ ({0, k 2 ,. .. , k 2 (n−1)}−Z +). This is also a necessary condition for the existence of positive Sonine formulas between the associated Bessel functions. In this paper we present two partial converse positive results: For k 1 ≥ 0, k 2 ∈ {1/2, 1, 2} and h > k 2 (n − 1), the operator V k ′ ,k is positive when restricted to functions which are invariant under the Weyl group, and there is an associated positive Sonine formula for the Bessel functions of type Bn. Moreover, the same positivity results hold for arbitrary k 1 ≥ 0, k 2 > 0 and h ∈ k 2 • Z +. The proof is based on a formula of Baker and Forrester on connection coefficients between multivariate Laguerre polynomials and an approximation of Bessel functions by Laguerre polynomials.
For a finite reflection group on R N , the associated Dunkl operators are parametrized firstorder... more For a finite reflection group on R N , the associated Dunkl operators are parametrized firstorder 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.
Transactions of the American Mathematical Society, Jun 17, 2015
The Heckman-Opdam hypergeometric functions of type BC extend classical Jacobi functions in one va... more The Heckman-Opdam hypergeometric functions of type BC extend classical Jacobi functions in one variable and include the spherical functions of non-compact Grassmann manifolds over the real, complex or quaternionic numbers. There are various limit transitions known for such hypergeometric functions. In the present paper, we use an explicit form of the Harish-Chandra integral representation as well as an interpolated variant in order to obtain two limit results, each of them for three continuous classes of hypergeometric functions of type BC which extend the group cases over the fields R, C, H. These limits are distinguished from the known results by explicit and uniform error bounds. The first limit realizes the approximation of the spherical functions of infinite dimensional Grassmannians of fixed rank; here hypergeometric functions of type A appear as limits. The second limit is a contraction limit towards Bessel functions of Dunkl type.
There exist several multivariate extensions of the classical Sonine integral representation for B... more There exist several multivariate extensions of the classical Sonine integral representation for Bessel functions of some index µ + ν with respect to such functions of lower index µ. For Bessel functions on matrix cones, Sonine formulas involve beta densities βµ,ν on the cone and trace already back to Herz. The Sonine representations known so far on symmetric cones are restricted to continuous ranges ℜµ, ℜν > µ 0 , where the involved beta densities are probability measures and the limiting index µ 0 ≥ 0 depends on the rank of the cone. It is zero only in the one-dimensional case, but larger than zero in all multivariate cases. In this paper, we study the extension of Sonine formulas for Bessel functions on symmetric cones to values of ν below the critical limit µ 0. This is achieved by an analytic extension of the involved beta measures as tempered distributions. Following recent ideas by A. Sokal for Riesz distributions on symmetric cones, we analyze for which indices the obtained beta distributions are still measures. At the same time, we characterize the indices for which a Sonine formula between the related Bessel functions exists. As for Riesz distributions, there occur gaps in the admissible range of indices which are determined by the so-called Wallach set.
Transactions of the American Mathematical Society, Jan 14, 2003
It is an open conjecture that generalized Bessel functions associated with root systems have a po... more 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.
The Heckman-Opdam hypergeometric functions of type BC extend classical Jacobi functions in one va... more The Heckman-Opdam hypergeometric functions of type BC extend classical Jacobi functions in one variable and include the spherical functions of non-compact Grassmann manifolds over the real, complex or quaternionic numbers. There are various limit transitions known for such hypergeometric functions, see e.g. [dJ], [RKV]. In the present paper, we use an explicit form of the Harish-Chandra integral representation as well as an interpolated variant, in order to obtain limit results for three continuous classes of hypergeometric functions of type BC which are distinguished by explicit, sharp and uniform error bounds. The first limit realizes the approximation of the spherical functions of infinite dimensional Grassmannians of fixed rank; here hypergeometric functions of type A appear as limits. The second limit is a contraction limit towards Bessel functions of Dunkl type.
In this paper we introduce probability-preserving convolution algebras on cones of positive semid... more In this paper we introduce probability-preserving convolution algebras on cones of positive semidefinite matrices over one of the division algebras F = R, C or H which interpolate the convolution algebras of radial bounded Borel measures on a matrix space M p,q (F) with p q. Radiality in this context means invariance under the action of the unitary group U p (F) from the left. We obtain a continuous series of commutative hypergroups whose characters are given by Bessel functions of matrix argument. Our results generalize well-known structures in the rank one case, namely the Bessel-Kingman hypergroups on the positive real line, to a higher rank setting. In a second part of the paper, we study structures depending only on the matrix spectra. Under the mapping r → spec(r), the convolutions on the underlying matrix cone induce a continuous series of hypergroup convolutions on a Weyl chamber of type B q. The characters are now Dunkl-type Bessel functions. These convolution algebras on the Weyl chamber naturally extend the harmonic analysis for Cartan motion groups associated with the Grassmann manifolds U (p, q)/(U p × U q) over F.
Dunkl operators are differential-difference operators on R N which generalize partial derivatives... more Dunkl operators are differential-difference operators on R N which generalize partial derivatives. They lead to generalizations of Laplace operators, Fourier transforms, heat semigroups, Hermite polynomials, and so on. In this paper we introduce two systems of biorthogonal polynomials with respect to Dunkl's Gaussian distributions in a quite canonical way. These systems, called Appell systems, admit many properties known from classical Hermite polynomials, and turn out to be useful for the analysis of Dunkl's Gaussian distributions. In particular, these polynomials lead to a new proof of a generalized formula of Macdonald due to Dunkl. The ideas for this paper are taken from recent works on non-Gaussian white noise analysis and from the umbral calculus.
It was recently shown by the authors that deformations of hypergroup convolutions w.r.t. positive... more It was recently shown by the authors that deformations of hypergroup convolutions w.r.t. positive semicharacters can be used to explain probabilistic connections between the Gelfand pairs (SL(d, C), SU (d)) and Hermitian matrices. We here study connections between general convolution semigroups on commutative hypergroups and their deformations. We are able to develop a satisfying theory, if the underlying positive semicharacter has some growth property. We present several examples which indicate that this growth condition holds in many interesting cases.
If G is a closed subgroup of a commutative hypergroup K, then the coset space K/G carries a quoti... more If G is a closed subgroup of a commutative hypergroup K, then the coset space K/G carries a quotient hypergroup structure. In this paper, we study related convolution structures on K/G coming from deformations of the quotient hypergroup structure by certain functions on K which we call partial characters with respect to G. They are usually not probability-preserving, but lead to so-called signed hypergroups on K/G. A first example is provided by the Laguerre convolution on [0, ∞[, which is interpreted as a signed quotient hypergroup convolution derived from the Heisenberg group. Moreover, signed hypergroups associated with the Gelfand pair U (n, 1), U (n) are discussed.
We study convolution algebras associated with Heckman-Opdam polynomials. For root systems of type... more We study convolution algebras associated with Heckman-Opdam polynomials. For root systems of type BC we derive three continuous classes of positive convolution algebras (hypergroups) by interpolating the double coset convolution structures of compact Grassmannians U/K with fixed rank over the real, complex or quaternionic numbers. These convolution algebras are linked to explicit positive product formulas for Heckman-Opdam polynomials of type BC, which occur for certain discrete multiplicities as the spherical functions of U/K. The results complement those of [21] for the non-compact case.
Proceedings of the American Mathematical Society, 1999
There exists a generalized Hankel transform of order α ≥ −1/2 on R, which is based on the eigenfu... more There exists a generalized Hankel transform of order α ≥ −1/2 on R, which is based on the eigenfunctions of the Dunkl operator Tαf (x) = f (x) + α + 1 2 f(x) − f(−x) x , f ∈ C 1 (R). For α = −1/2 this transform coincides with the usual Fourier transform on R. In this paper the operator Tα replaces the usual first derivative in order to obtain a sharp uncertainty principle for generalized Hankel transforms on R. It generalizes the classical Weyl-Heisenberg uncertainty principle for the position and momentum operators on L 2 (R); moreover, it implies a Weyl-Heisenberg inequality for the classical Hankel transform of arbitrary order α ≥ −1/2 on [0, ∞[.
We present an explicit product formula for the spherical functions of the compact Gelfand pairs (... more We present an explicit product formula for the spherical functions of the compact Gelfand pairs (G, K 1) = (SU (p + q), SU (p) × SU (q)) with p ≥ 2q, which can be considered as the elementary spherical functions of one-dimensional K-type for the Hermitian symmetric spaces G/K with K = S(U (p) × U (q)). Due to results of Heckman, they can be expressed in terms of Heckman-Opdam Jacobi polynomials of type BC q with specific half-integer multiplicities. By analytic continuation with respect to the multiplicity parameters we obtain positive product formulas for the extensions of these spherical functions as well as associated compact and commutative hypergroup structures parametrized by real p ∈]2q − 1, ∞[. We also obtain explicit product formulas for the involved continuous two-parameter family of Heckman-Opdam Jacobi polynomials with regular, but not necessarily positive multiplicities. The results of this paper extend well known results for the disk convolutions for q = 1 to higher rank.
Stochastic Processes and their Applications, Aug 1, 1991
Optimal or asymptotically optimal linear unbiased mean estimators for a wide class of weakly stat... more Optimal or asymptotically optimal linear unbiased mean estimators for a wide class of weakly stationary processes including ARMA processes are derived explicitly. Furthermore their convergence rates are given.
In this paper, we derive explicit product formulas and positive convolution structures for three ... more In this paper, we derive explicit product formulas and positive convolution structures for three continuous classes of Heckman-Opdam hypergeometric functions of type BC. For specific discrete series of multiplicities these hypergeometric functions occur as the spherical functions of non-compact Grassmann manifolds G/K over one of the skew fields F = R, C, H. We write the product formula of these spherical functions in an explicit form which allows analytic continuation with respect to the parameters. In each of the three cases, we obtain a series of hypergroup algebras which include the commutative convolution algebras of K-biinvariant functions on G as special cases. The characters are given by the associated hypergeometric functions.
These lecture notes are intended as an introduction to the theory of rational Dunkl operators and... more 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.
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