Sampling in Spaces of Bandlimited Functions on Commutative Spaces

Abstract

A homogeneous space $\mathbf{X} = G/K$ is called commutative if G is a locally compact group, K is a compact subgroup, and the Banach ∗ -algebra ${L}^{1}{(\mathbf{X})}^{K}$ of K-invariant integrable functions on $\mathbf{X}$ is commutative. In this chapter we introduce the space ${L}_{\Omega }^{2}(\mathbf{X})$ of $\Omega$-bandlimited function on $\mathbf{X}$ by using the spectral decomposition of ${L}^{2}(\mathbf{X})$. 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 ${L}_{\Omega }^{2}(\mathbf{X})$. At the end we discuss the example of ${\mathbb{R}}^{d}$, the spheres S ^d, compact symmetric spaces, and the Heisenberg group realized as the commutative space $\mathrm{U}(n) \ltimes {\mathbb{H}}_{n}/\mathrm{U}(n)$.