Fourier analysis on domains in compact groups

Let Ω be a measurable subset of a compact group G of positive Haar measure. Let μ:π →μπ be a non-negative function defined on the dual space G and let L2(μ) be the corresponding Hilbert space which consists of elements (ξπ )π∈suppμ satisfying μπ Tr(ξπ ξ ∗π) < ∞, where ξπ is a linear operator on the representation space of π, and is equipped with the inner product: ((ξπ ), (ηπ )) = μπ Tr(ξπ η∗π ). We show that the Fourier transform gives an isometric isomorphism from L2(Ω) onto L2(μ) if and only if the restrictions to Ω of all matrix coordinate functions √μππij , π ∈ suppμ, constitute an orthonormal basis for L2(Ω). Finally compact connected Lie groups case is studied. © 2006 Elsevier Inc. All rights reserved.