Operator-Valued Continuous Gabor Transforms over Non-unimodular Locally Compact Groups

In this article, we present the abstract harmonic analysis aspects of the operator-valued continuous Gabor transform (CGT) on second countable, non-unimodular, and type I locally compact groups. We show that the operator-valued continuous Gabor transform CGT satisfies a Plancherel formula and an inversion formula. As an example, we study these results on the continuous affine group

Square-integrability of multivariate metaplectic wave-packet representations

This paper presents a systematic study for harmonic analysis of metaplectic wave-packet representations on the Hilbert function space L2(Rd). The abstract notions of symplectic wave-packet groups and metaplectic wave-packet representations will be introduced. We then present an admissibility condition on closed subgroups of the real symplectic group Sp(Rd), which guarantees the square-integrability of the associated metaplectic wave-packet representation on L2(Rd)

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

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 μ 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 L1(G/H,μ). 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

Abstract Poisson summation formulas over homogeneous spaces of compact groups

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

Abstract Plancherel (Trace) Formulas over Homogeneous Spaces of Compact Groups

This paper introduces a unified operator theory approach to the abstract Plancherel (trace) formulas over homogeneous spaces of compact groups. Let G be a compact group and let H be a closed subgroup of G . Let G/H be the left coset space of H in G and let μ be the normalized G -invariant measure on G/H associated with Weil’s formula. Then we present a generalized abstract notion of Plancherel (trace) formula for the Hilbert space L2(G/H,μ)

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

Generalized wavelet transforms over finite fields

In this article we introduce the abstract notion of generalized wavelet (affine) groups over finite fields as the finite group of generalized dilations, and translations. We then present a unified theoretical linear algebra approach to the theory of generalized wavelet transforms over finite fields. It is shown that each vector defined over a finite field can be represented as a finite coherent sum of generalized wavelet coefficients as well

A Class of Abstract Linear Representations for Convolution Function Algebras over Homogeneous Spaces of Compact Groups

This paper introduces a class of abstract linear representations on Banach convolution function algebras over homogeneous spaces of compact groups. Let G be a compact group and H a closed subgroup of G . Let μ be the normalized G -invariant measure over the compact homogeneous space G/H associated with Weil's formula and 1≤p<∞ . We then present a structured class of abstract linear representations of the Banach convolution function algebras Lp(G/H,μ)

Square-integrability of metaplectic wave-packet representations onL2(R)

This paper presents a systematic study for harmonic analysis of metaplectic wave-packets in L2(R) via group representation theory. The abstract notions of metaplectic wave-packet groups and metaplectic wave-packet representations will be presented and as the main result, we prove an admissibility condition on closed subgroups of the special linear group SL(2,R), which guarantees square integrability of the associated metaplectic wave-packet representation on L2(R). Finally, we present an analytic study of metaplectic wave-packet representations over admissible subgroups of SL(2,R)