The Composite Cosine Transform on the Stiefel Manifold and Generalized Zeta Integrals

2005/01/02 by E. Ournycheva, Ournycheva, E., B. Rubin +1
Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #Primary 42B10 #Secondary 52A22 #math.FA #msc:42B10 #msc:52A22

paper · pdf · doi:10.48550/arxiv.math/0501020

21 pages; updated Introduction, Sec. 4.1, and Sec. 5, added references

arxiv created 2005/03/08 · arxiv updated 2009/12/01

Abstract

We introduce a new integral transform T^\lam f, \lam ∈ C^m, on the Stiefel manifold of orthonormal m-frames in Rⁿ which generalizes the \lam-cosine transform on the Grassmann manifold of m-dimensional linear subspaces of Rⁿ. We call it the composite cosine transform, by taking into account that its kernel agrees with the composite power function of the cone of positive definite symmetric matrices. Our aim is to describe the set of all \lam ∈ C^m for which T^\lam is injective on the space of integrable functions. We obtain the precise description of this set in some important cases, in particular, for \lam-cosine transforms on Grassmann manifolds. The main tools are the classical Fourier analysis of functions of matrix argument and the relevant zeta integrals.

Related

• Composite Cosine Transforms
• Analytic and Group-Theoretic Aspects of the Cosine Transform
• Funk, Cosine, and Sine Transforms on Stiefel and Grassmann manifolds, II
• The λ-Cosine Transforms, Differential Operators, and Funk Transforms on Stiefel and Grassmann Manifolds
• Funk, Cosine, and Sine Transforms on Stiefel and Grassmann manifolds