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A Characterization of Metaplectic Time-Frequency Representations

Karlheinz Gröchenig, Irina Shafkulovska

TL;DR

This work classifies all time-frequency representations that satisfy a general covariance property, showing that any nonzero bilinear, separately weak*-continuous mapping with a measurable intertwining function must be a scalar multiple of a metaplectic Wigner distribution, ${\mathfrak{R}}(f,g) = a\, {\hat{\mathcal{A}}}(f\otimes g)$ with ${\mathcal{A}} \in {\mathrm{Sp}}(4d,{\mathbb{R}})$. The authors build the argument by combining a bilinear, weak*-continuous kernel representation, a Schur-type commutation analysis, and Lie group techniques to prove that the intertwining map ${\Phi}$ is linear and symplectic. Consequently, metaplectic representations arise intrinsically as the only covariant time-frequency representations under the stated hypotheses, linking to Cohen's class and pseudodifferential operator analysis. The results illuminate the geometric and algebraic structure underlying time-frequency analysis and justify the central role of metaplectic operators in this domain.

Abstract

We characterize all time-frequency representations that satisfy a general covariance property: any weak*-continuous bilinear mapping that intertwines time-frequency shifts on the configuration space with time-frequency shifts on phase space is a multiple of a metaplectic time-frequency representation.

A Characterization of Metaplectic Time-Frequency Representations

TL;DR

This work classifies all time-frequency representations that satisfy a general covariance property, showing that any nonzero bilinear, separately weak*-continuous mapping with a measurable intertwining function must be a scalar multiple of a metaplectic Wigner distribution, with . The authors build the argument by combining a bilinear, weak*-continuous kernel representation, a Schur-type commutation analysis, and Lie group techniques to prove that the intertwining map is linear and symplectic. Consequently, metaplectic representations arise intrinsically as the only covariant time-frequency representations under the stated hypotheses, linking to Cohen's class and pseudodifferential operator analysis. The results illuminate the geometric and algebraic structure underlying time-frequency analysis and justify the central role of metaplectic operators in this domain.

Abstract

We characterize all time-frequency representations that satisfy a general covariance property: any weak*-continuous bilinear mapping that intertwines time-frequency shifts on the configuration space with time-frequency shifts on phase space is a multiple of a metaplectic time-frequency representation.
Paper Structure (5 sections, 11 theorems, 53 equations)

This paper contains 5 sections, 11 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Symplectic matrices and metaplectic operators
  3. Analytic consequences
  4. Consequences of the General Covariance Property
  5. Final Remarks

Key Result

Theorem 1.1

Let ${\mathfrak{R}}:{\mathcal{S}({{\mathbb R}^d})}\times {\mathcal{S}({{\mathbb R}^d})}\to {\mathcal{S}'({{\mathbb R}^{2d}})}$ be a non-zero, bilinear, separately weak*-continuous mapping satisfying the general covariance property eq:vertauschung with a measurable function $\Phi$. Then there exist

Theorems & Definitions (22)

  • Theorem 1.1: Main result
  • Corollary 1.2
  • Remark 1.3
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • ...and 12 more