Absolutely Convergent Fourier Series of Functions over Homogeneous Spaces of Compact Groups

This paper presents a systematic study for classical aspects of functions with absolutely convergent Fourier series over homogeneous spaces of compact groups. Let G be a compact group, H be a closed subgroup of G, and μ be the normalized G-invariant measure over the left coset space G/H associated with Weil’s formula with respect to the probability measures of G and H. We introduce the abstract notion of functions with absolutely convergent Fourier series in the Banach function space L1(G/H,μ). We then present some analytic characterizations for the linear space consisting of functions with absolutely convergent Fourier series over the compact homogeneous space G/H.