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Composite Cosine Transforms

Part of: Harmonic analysis in several variables General convexity

Published online by Cambridge University Press:  21 December 2009

E. Ournycheva  and
B. Rubin
E. Ournycheva
Affiliation:
Department of Mathematical Sciences, Kent State University, Mathematics and Computer Science Building, Summit Street, Kent OH 44242, U.S.A. E-mail: ournyche@math.kent.edu
B. Rubin
Affiliation:
Department of Mathematics, Louisiana State University, Baton Rouge, LA, 70803, U.S.A. and Institute of Mathematics, Hebrew University, Jerusalem 91904, Israel. E-mail: borisr@math.lsu.edu
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Abstract

The cosine transforms of functions on the unit sphere play an important role in convex geometry, Banach space theory, stochastic geometry and other areas. Their higher-rank generalization to Grassmann manifolds represents an interesting mathematical object useful for applications. More general integral transforms are introduced that reveal distinctive features of higher-rank objects in full generality. These new transforms are called the composite cosine transforms, by taking into account that their kernels agree with the composite power function of the cone of positive definite symmetric matrices. It is shown that injectivity of the composite cosine transforms can be studied using standard tools of the Fourier analysis on matrix spaces. In the framework of this approach, associated generalized zeta integrals are introduced and new simple proofs given to the relevant functional relations. The technique is based on application of the higher-rank Radon transform on matrix spaces.

MSC classification

Secondary: 42B10: Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type 52A22: Random convex sets and integral geometry

Information

Type
Research Article
Information
Mathematika , Volume 52 , Issue 1-2 , December 2005 , pp. 53 - 68
Copyright
Copyright © University College London 2005

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