Trigonometric polynomials over homogeneous spaces of compact groups

Abstract

This paper presents a systematic study for trigonometric polynomials over homogeneous spaces of compact groups. Let $H$ be a closed subgroup of a compact group $G$. Using the abstract notion of dual space $\widehat{G/H}$, we introduce the space of trigonometric polynomials $\mathrm{Trig}(G/H)$ over the compact homogeneous space $G/H$. As an application for harmonic analysis of trigonometric polynomials, we prove that the abstract dual space of anyhomogeneous space of compact groups separates points of the homogeneous space in some sense.