Abstract Poisson summation formulas over homogeneous spaces of compact groups

Abstract

Poisson summation formulas over homogeneous spaces of compact groupsAbstract This paper presents the abstract notion of Poisson summation formulas for 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 to the Weil's formula. We prove that the abstract Fourier transform over G/H satisfies a generalized version of the Poisson summation formula.A. G. Farashahi groups of compact non-Abelian groups is placed as building blocks for coherent states analysis, constructive approximation, and particle physics, see [4,10,11,[15][16][17][18][19] and references therein. Over the last decades, abstract and computational aspects of Fourier transforms and Plancherel formulas over symmetric spaces have achieved significant popularity in geometric analysis, mathematical physics, and scientific computing, see [6,8,9] and references therein.Let G be a compact group and H be a closed subgroup of G. Then the left coset space G/H is a compact homogeneous space, where G acts on it via the left action. Let μ be the normalized G-invariant measure on the compact homogeneous space G/H associated to the Weil's formula with respect to the probability measures of G and H . This article consists of theoretical aspects of a unified approach to the nature of abstract Poisson summation formulas over homogeneous spaces of compact groups.This paper is organized as follows. Section 2 is devoted to fixing notation and preliminaries including a summary on non-Abelian Fourier analysis over compact groups and classical results on abstract harmonic analysis over compact homogeneous spaces. In Sect. 3, we present some abstract aspects of harmonic analysis on the Banach functions spaces over the compact homogeneous space G/H . We then introduce the abstract notion of dual space G/H for the homogeneous space G/H . Next we present the theoretical definition of the abstract operator-valued Fourier transform over the compact homogeneous space G/H . The paper is concluded by presenting an abstract generalized version of the Poisson summation formula. Preliminaries and notationLet H be a separable Hilbert space. An operator T ∈ B(H) is called a Hilbert-Schmidt operator if for one, hence for any orthonormal basis {e k } of H we have k T e k 2 < ∞. The set of all Hilbert-Schmidt operators on H is denoted by HS(H) and for T ∈ HS(H) the Hilbert-Schmidt norm of T is T 2 HS = k T e k 2 . The set HS(H) is a self adjoint two sided ideal in B(H) and if H is finite-dimensional we have HS(H) = B(H). An operator T ∈ B(H) is trace-class, whenever T tr = tr[|T |] < ∞, if tr[T ] = k T e k , e k and |T | = (T T * )

Abstract Structure of Measure Algebras on Coset Spaces of Compact Subgroups in Locally Compact Groups

“…Over the last decades, some new aspects and applications of Banach convolution algebras have achieved significant popularity in different areas such as constructive approximation [4,5,6], and theoretical aspects of coherent state (covariant) analysis, see [26] and references therein. Homogeneous spaces are group-like structures with many applications in mathematical physics, differential geometry, geometric analysis, and coherent state (covariant) transforms, see [8,14,19,27,28,29].…”

Noncommutative Fourier Series on $\mathbb{Z}^2\backslash SE(2)$

“…In a nutshell, homogeneous spaces are group-like structures with many interesting applications in mathematical physics, differential geometry, geometric analysis, and coherent state (covariant) transforms, see [11,25,30]. Fourier expansions on homogeneous spaces of compact groups, coset spaces of compact groups, have been studied at depth in [14,15,16]. This theory is strongly benefited from the compactness assumption about the group which is not as the case for SE (2) and hence a different approach is required for the right coset space Z 2 \SE (2).…”