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THE PALEYWIENER THEOREM AND THE LOCAL HUYGENS ’ PRINCIPLE FOR COMPACT SYMMETRIC SPACES
, 2004
"... Dedicated to Gerrit van Dijk on the occasion of his 65th birthday Abstract. We prove a PaleyWiener Theorem for a class of symmetric spaces of the compact type, in which all root multiplicities are even. This theorem characterizes functions of small support in terms of holomorphic extendability and ..."
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Dedicated to Gerrit van Dijk on the occasion of his 65th birthday Abstract. We prove a PaleyWiener Theorem for a class of symmetric spaces of the compact type, in which all root multiplicities are even. This theorem characterizes functions of small support in terms of holomorphic extendability and exponential type of their (discrete) Fourier transforms. We also provide three independent new proofs of the strong Huygens ’ principle for a suitable constant shift of the wave equation on odddimensional spaces from our class.
Huygens’ principle in Minkowski spaces and soliton solutions of the Kortewegde Vries equation
 Comm. Math. Phys
, 1997
"... A new class of linear second order hyperbolic partial differential operators satisfying Huygens ’ principle in Minkowski spaces is presented. The construction reveals a direct connection between Huygens ’ principle and the theory of solitary wave solutions of the Kortewegde Vries equation. Mathemat ..."
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A new class of linear second order hyperbolic partial differential operators satisfying Huygens ’ principle in Minkowski spaces is presented. The construction reveals a direct connection between Huygens ’ principle and the theory of solitary wave solutions of the Kortewegde Vries equation. Mathematics Subject Classification: 35Q51, 35Q53, 35L05, 35L15, 35Q05.
Radiation fields, scattering and inverse scattering on asymptotically hyperbolic manifolds
, 2004
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Equipartition Of Energy For Waves In Symmetric Space
 J. Funct. Anal
, 1991
"... . Let X = G=K be an odddimensional semisimple Riemannian symmetric space of the noncompact type, and suppose that all Cartan subgroups of G are conjugate. Let u be a realvalued classical solution of the modified wave equation u tt = (\Delta + k)u on R \Theta X, the Cauchy data of which are support ..."
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Cited by 7 (2 self)
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. Let X = G=K be an odddimensional semisimple Riemannian symmetric space of the noncompact type, and suppose that all Cartan subgroups of G are conjugate. Let u be a realvalued classical solution of the modified wave equation u tt = (\Delta + k)u on R \Theta X, the Cauchy data of which are supported in a closed metric ball of radius a at time t = 0. Here t is the coordinate on R, \Delta is the (nonpositive definite) LaplaceBeltrami operator on X, and k is a positive constant depending on the root structure of the Lie algebra of G. We show that the (tindependent) energy functional of u is eventually (for jtj a) partitioned into equal potential and kinetic parts; specifically, half the integrals over X of u 2 t and jduj 2 \Gamma ku 2 respectively, where d is the exterior derivative in X. The proof uses Helgason's PaleyWiener Theorem for X, the classical PaleyWiener Theorem, and properties of HarishChandra's c function. 1. Introduction and statement of the theorem. The ce...
Normally Hyperbolic Operators, the Huygens Property and Conformal Geometry
, 1996
"... In this paper we give a review on normally hyperbolic operators of Huygens type. The methods to determine Huygens operators we explain here were essentially influenced and developed by Paul Gunther. Contents 1 Introduction 2 2 Riesz distributions 4 2.1 Riesz distributions on Minkowski spaces : : : ..."
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In this paper we give a review on normally hyperbolic operators of Huygens type. The methods to determine Huygens operators we explain here were essentially influenced and developed by Paul Gunther. Contents 1 Introduction 2 2 Riesz distributions 4 2.1 Riesz distributions on Minkowski spaces : : : : : : : : : : : : : : : : : : 4 2.2 Riesz distributions on geodesically normal domains in Lorentzian manifolds : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 3 Normally hyperbolic operators 8 3.1 Definition and examples : : : : : : : : : : : : : : : : : : : : : : : : : : : 8 3.2 The Weitzenbock formula : : : : : : : : : : : : : : : : : : : : : : : : : : 9 3.3 Hadamard coefficients : : : : : : : : : : : : : : : : : : : : : : : : : : : : 10 3.4 Fundamental solution : : : : : : : : : : : : : : : : : : : : : : : : : : : : 12 4 Huygens operators 14 5 Conformal gauge invariance of Huygens operators 15 6 The moments of normally hyperbolic operators 18 6.1 Mappings with simple tran...
Helmholtz Operators And Symmetric Space Duality
 Invent. Math
, 1997
"... . We consider the property of vanishing logarithmic term (VLT) for the fundamental solution of the shifted Laplaced'Alembert operators + b (b a constant), on pseudoRiemannian reductive symmetric spaces M . Our main result is that such an operator on the cdual or FlenstedJensen dual of M has ..."
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. We consider the property of vanishing logarithmic term (VLT) for the fundamental solution of the shifted Laplaced'Alembert operators + b (b a constant), on pseudoRiemannian reductive symmetric spaces M . Our main result is that such an operator on the cdual or FlenstedJensen dual of M has the VLT property if and only if a corresponding operator on M does. For Lorentzian spaces, where the + b are hyperbolic, VLT is known to be equivalent to the strong Huygens principle. We use our results to construct a large supply of new (space, operator) pairs satisfying Huygens' principle. 1. Introduction and principal results. An old theme in geometry is that of partial differential equations on manifolds which have particularly nice fundamental solutions. For hyperbolic equations, one might want sharp, i.e. dispersion free, propagation of sharp signals. This property, Huygens' principle, can be detected by studying the asymptotics of the fundamental solution, in the vanishing of certain ter...
WAVE PHENOMENA, HUYGENS ' PRINCIPLE AND 3D ELASTIC WAVES
"... 117 We briefly discuss some important results on Huygens ' principle in the sense of Hadamard's minor premise. We also indicate a negative result we obtained concerning the above principle and the system of elastic waves in the presence of an impurity of the medium. We also mention some op ..."
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117 We briefly discuss some important results on Huygens ' principle in the sense of Hadamard's minor premise. We also indicate a negative result we obtained concerning the above principle and the system of elastic waves in the presence of an impurity of the medium. We also mention some open problems in the subject as well as possible interesting generalizations. 1.