Groups, Geometry, and Dynamics


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Volume 11, Issue 4, 2017, pp. 1437–1467
DOI: 10.4171/GGD/434

Published online: 2017-12-07

Abstract operator-valued Fourier transforms over homogeneous spaces of compact groups

Arash Ghaani Farashahi[1]

(1) University of Vienna, Austria

This paper presents a systematic theoretical study for the abstract notion of operator-valued Fourier transforms 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 to the Weil's formula. We introduce the generalized notions of abstract dual homogeneous space $\widehat{G/H}$ for the compact homogeneous space $G/H$ and also the operator-valued Fourier transform over the Banach function space $L^1(G/H,\mu)$. We prove that the abstract Fourier transform over $G/H$ satisfies the Plancherel formula and the Poisson summation formula.

Keywords: Compact group, homogeneous space, coset space,Weil’s formula, dual homogeneous space, trigonometric polynomial, Fourier transform, Plancherel (trace) formula, Hausdorff–Young inequality, inversion formula, Poisson summation formula

Ghaani Farashahi Arash: Abstract operator-valued Fourier transforms over homogeneous spaces of compact groups. Groups Geom. Dyn. 11 (2017), 1437-1467. doi: 10.4171/GGD/434