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A homogeneous space X=G/K is called commutative if G is a locally compact group, K is a compact subgroup, and the Banach ∗ -algebra L1(X)K of K-invariant integrable functions on X is commutative. In this chapter we introduce the space L2Ω(X) of Ω-bandlimited function on X by using the spectral decomposition of L2(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 L2Ω(X). At the end we discuss the example of ℝ, the spheres S d, compact symmetric spaces, and the Heisenberg group realized as the commutative space (formula presented).

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Applied and Numerical Harmonic Analysis

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