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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
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.
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.
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.