Sampling in Spaces of Bandlimited Functions on Commutative Spaces

TL;DR

This paper introduces and analyzes A-bandlimited functions on commutative homogeneous spaces, establishing their structure as reproducing kernel Hilbert spaces and deriving sampling theorems based on smoothness properties.

Contribution

It defines A-bandlimited function spaces on commutative spaces, characterizes their reproducing kernels, and proves sampling results using smoothness, extending classical sampling theory.

Findings

Spaces are reproducing kernel Hilbert spaces with explicitly determined kernels.

Sampling theorems are established based on the smoothness of functions.

Applications include Euclidean spaces, spheres, symmetric spaces, and the Heisenberg group.

Abstract

A connected homogeneous space X=G/K is called commutative if G is a connected Lie group, $K$ 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).