Sampling in Spaces of Bandlimited Functions on Commutative Spaces
Functional Analysis
2011-07-25 v1
Abstract
A connected homogeneous space X=G/K is called commutative if G is a connected Lie group, is a compact subgroup and the B*-algebra L^1(X)^K of K-invariant integrable function on X is commutative. In this article we introduce the space L^2_A (X) of A-bandlimited function on X by using the spectral decomposition of L^2 (X). We show that those spaces are reproducing kernel Hilbert spaces and determine the reproducing kernel. We then prove sampling results for those spaces using the smoothness of the elements in L^2_A (X). At the end we discuss the example of R^d, the spheres S^d, compact symmetric spaces and the Heisenberg group realized as the commutative space U (n) x H_n/U (n).
Keywords
Cite
@article{arxiv.1107.4578,
title = {Sampling in Spaces of Bandlimited Functions on Commutative Spaces},
author = {Jens Gerlach Christensen and Gestur Olafsson},
journal= {arXiv preprint arXiv:1107.4578},
year = {2011}
}