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ELEMENTARY PROOFS OF PALEY–WIENER THEOREMS FOR THE DUNKL TRANSFORM ON THE REAL LINE
, 2005
"... Abstract. We give an elementary proof of the Paley–Wiener theorem for smooth functions for the Dunkl transforms on the real line, establish a similar theorem for L 2functions and prove identities in the spirit of Bang for L pfunctions. The proofs seem to be new also in the special case of the Four ..."
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Abstract. We give an elementary proof of the Paley–Wiener theorem for smooth functions for the Dunkl transforms on the real line, establish a similar theorem for L 2functions and prove identities in the spirit of Bang for L pfunctions. The proofs seem to be new also in the special case of the Fourier transform. 1. Introduction and
Integral representation and sharp asymptotic results for some HeckmanOpdam hypergeometric functions of type BC
, 2014
"... The HeckmanOpdam hypergeometric functions of type BC extend classical Jacobi functions in one variable and include the spherical functions of noncompact Grassmann manifolds over the real, complex or quaternionic numbers. There are various limit transitions known for such hypergeometric functions, ..."
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The HeckmanOpdam hypergeometric functions of type BC extend classical Jacobi functions in one variable and include the spherical functions of noncompact 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 HarishChandra 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.
SUPPORT PROPERTIES AND HOLMGREN’S UNIQUENESS THEOREM FOR DIFFERENTIAL OPERATORS WITH HYPERPLANE SINGULARITIES
, 2004
"... Abstract. Let W be a finite Coxeter group acting linearly on R n. In this article we study support properties of Winvariant partial differential operator D on R n with real analytic coefficients. Our assumption is that the principal symbol of D has a special form, related to the root system corresp ..."
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Abstract. Let W be a finite Coxeter group acting linearly on R n. In this article we study support properties of Winvariant partial differential operator D on R n with real analytic coefficients. Our assumption is that the principal symbol of D has a special form, related to the root system corresponding to W. In particular the zeros of the principal symbol are supposed to be located on hyperplanes fixed by reflections in W. We show that conv(suppDf) = conv(supp f) holds for all compactly supported smooth functions f so that conv(suppf) is Winvariant. The main tools in the proof are Holmgren’s uniqueness theorem and some elementary convex geometry. Several examples and applications linked to the theory of special functions associated with root systems are presented.
On Fourier and Hankel Sampling
"... We use the Paley–Wiener theorem for the Fourier and Hankel transforms to compare Fourier and Hankel Sampling. 1 ..."
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We use the Paley–Wiener theorem for the Fourier and Hankel transforms to compare Fourier and Hankel Sampling. 1
THREE RESULTS IN DUNKL ANALYSIS
, 904
"... a distinguished polish mathematician, a guide and a friend, who has left many orphans in Wroclaw and around the world. We miss you. Abstract. In this article, we establish first a geometric Paley–Wiener theorem for the Dunkl transform in the crystallographic case. Next we obtain an optimal bound for ..."
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a distinguished polish mathematician, a guide and a friend, who has left many orphans in Wroclaw and around the world. We miss you. Abstract. In this article, we establish first a geometric Paley–Wiener theorem for the Dunkl transform in the crystallographic case. Next we obtain an optimal bound for the L p → L p norm of Dunkl translations in dimension 1. Finally we describe more precisely the support of the distribution associated to Dunkl translations in higher dimension. 1.
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, 2014
"... A central limit theorem for random walks on the dual of a compact Grassmannian ..."
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A central limit theorem for random walks on the dual of a compact Grassmannian
A Central Limit Theorem for Random Walks on the Dual of a Compact Grassmannian
"... Abstract. We consider compact Grassmann manifolds G/K over the real, complex or quaternionic numbers whose spherical functions are Heckman–Opdam polynomials of type BC. From an explicit integral representation of these polynomials we deduce a sharp Mehler–Heine formula, that is an approximation of t ..."
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Abstract. We consider compact Grassmann manifolds G/K over the real, complex or quaternionic numbers whose spherical functions are Heckman–Opdam polynomials of type BC. From an explicit integral representation of these polynomials we deduce a sharp Mehler–Heine formula, that is an approximation of the Heckman–Opdam polynomials in terms of Bessel functions, with a precise estimate on the error term. This result is used to derive a central limit theorem for random walks on the semilattice parametrizing the dual of G/K, which are constructed by successive decompositions of tensor powers of spherical representations of G. The limit is the distribution of a Laguerre ensemble in random matrix theory. Most results of this paper are established for a larger continuous set of multiplicity parameters beyond the group cases.
SOME REMARKS ON ANOTHER PROOF OF GEOMETRICAL PALEY–WIENER THEOREMS FOR THE DUNKL TRANSFORM
, 2004
"... Abstract. We argue that another proof by Trimèche of the geometrical form of the Paley–Wiener theorems for the Dunkl transform is not correct. 1. ..."
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Abstract. We argue that another proof by Trimèche of the geometrical form of the Paley–Wiener theorems for the Dunkl transform is not correct. 1.