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The Radon transform of functions of matrix argument
E. Ournycheva,Boris Rubin +1 more
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In this paper, a systematic treatment of problems in analysis and integral geometry related to inversion of the Radon transform on the space of real rectangular matrices is presented, with a special emphasis on new higher rank phenomena, in particular, on possibly minimal conditions under which the radon transform is well defined and can be explicitly inverted.Abstract:
The monograph contains a systematic treatment of a circle of problems in analysis and integral geometry related to inversion of the Radon transform on the space of real rectangular matrices. This transform assigns to a function $f$ on the matrix space the integrals of $f$ over the so-called matrix planes, the linear manifolds determined by the corresponding matrix equations. Different inversion methods are discussed. They rely on close connection between the Radon transform, the Fourier transform, the Garding-Gindikin fractional integrals, and matrix modifications of the Riesz potentials. A special emphasis is made on new higher rank phenomena, in particular, on possibly minimal conditions under which the Radon transform is well defined and can be explicitly inverted. Apart of the space of Schwartz functions, we also employ $L^p$-spaces and the space of continuous functions. Many classical results for the Radon transform on $R^n$ are generalized to the higher rank case.read more
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Journal ArticleDOI
The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)
Journal ArticleDOI
A COURSE IN CONVEXITY (Graduate Studies in Mathematics 54) By ALEXANDER BARVINOK: 366 pp., US$59.00, ISBN 0-8218-2968-8 (American Mathematical Society, Providence, RI, 2002)
Journal ArticleDOI
Riesz potentials and integral geometry in the space of rectangular matrices
Boris Rubin,Boris Rubin +1 more
TL;DR: In this paper, the Fourier transform of Riesz potentials on the space of rectangular n × m matrices has been studied in the context of higher rank problems of harmonic analysis, representation theory, and integral geometry.
References
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