Trigonometric polynomials over homogeneous spaces of compact groups

This paper presents a systematic study for trigonometric polynomials over homogeneous spaces of compact groups. Let HH be a closed subgroup of a compact group GG. Using the abstract notion of dual space G/HˆG/H^, we introduce the space of trigonometric polynomials Trig(G/H)Trig(G/H) over the compact homogeneous space G/HG/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.