Abstract
This paper presents a systematic study for harmonic analysis of metaplectic wave-packet representations on the Hilbert function space . 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 , which guarantees the square-integrability of the associated metaplectic wave-packet representation on .
Highlights
Many intresting applications of mathematical analysis in theoretical physics prompt particular forms of multivariate metaplectic (Shale-Weyl) representation [14,15,16, 25, 41] under various names; quadratic-phase transforms, linear canonical transforms [10, 36], Fresnel transforms, fractional Fourier transforms [54], Gaussian integral [51]
This paper presents a systematic study for harmonic analysis of metaplectic wave-packet representations on the Hilbert function space L2(Rd)
We 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)
Summary
Many intresting applications of mathematical analysis in theoretical physics (e.g. paraxial optic, quantum mechanics, etc) prompt particular forms of multivariate metaplectic (Shale-Weyl) representation [14,15,16, 25, 41] under various names; quadratic-phase transforms, linear canonical transforms [10, 36], Fresnel transforms, fractional Fourier transforms [54], Gaussian integral [51]. From harmonic and functional analysis aspects such coherent structures are classically originated from squar-integrable representations of locally compact groups, see [33, 46, 50, 59] and references therein. The mathematical theory of classical wave-packet analysis on the real line is originated from classical dilations, translations, and modulations of a given window function. The mathematical theory of wave-packet analysis as a coherent state analysis has been recently abstracted in the setting of locally compact Abelian groups in [28]. We shall address analytic aspects of metaplectic wave-packet transforms over L2(Rd) using tools from representation theory of locally compact groups and abstract harmonic analysis. As an application of our results we study analytic aspects of metaplectic wave-packet transforms associated to closed subgroups of the real symplectic goup Sp(Rd). We will illustrate application of these techniques in the case of well-known compact subgroups of the real symplectic group Sp(Rd)
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