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

Abstract

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 $\mu$ 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 $L^1(G/H,\mu )$. We then present some analytic characterizations for the linear space consisting of functions with absolutely convergent Fourier series over the compact homogeneous space $G/H$.