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Metaplectic Quantum Time--Frequency Analysis, Operator Reconstruction and Identification

Henry McNulty

Abstract

The problem of identifying and reconstructing operators from a diagonal of the Gabor matrix is considered. The framework of Quantum Time--Frequency Analysis is used, wherein this problem is equivalent to the discretisation of the diagonal of the polarised Cohen's class of the operator. Metaplectic geometry allows the generalisation of conditions on appropriate operators, giving sets of operators which can be reconstructed and identified on the diagonal of the discretised polarised Cohen's class of the operator.

Metaplectic Quantum Time--Frequency Analysis, Operator Reconstruction and Identification

Abstract

The problem of identifying and reconstructing operators from a diagonal of the Gabor matrix is considered. The framework of Quantum Time--Frequency Analysis is used, wherein this problem is equivalent to the discretisation of the diagonal of the polarised Cohen's class of the operator. Metaplectic geometry allows the generalisation of conditions on appropriate operators, giving sets of operators which can be reconstructed and identified on the diagonal of the discretised polarised Cohen's class of the operator.

Paper Structure

This paper contains 10 sections, 21 theorems, 170 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Harmonic and Time--Frequency Analysis
  4. Gabor Frames
  5. Quantisation Schemes
  6. Quantum Time--Frequency Analysis
  7. Symplectic Matrices and Metapletic Operators
  8. Discrete Reconstruction of Underspread Operators
  9. Metaplectic Wigner distributions and operator reconstruction
  10. From Reconstruction to Identification

Key Result

Theorem 1.1

Given a lattice $\Lambda \subset \mathbb{R}^{2d}$ with adjoint lattice $\Lambda^\circ = A\mathbb{Z}^{2d}$, let $Q$ denote the fundamental domain of $\Lambda^\circ$ given by $Q=A[-\tfrac{1}{2},\tfrac{1}{2}]^{2d}$. Let $h\in C^{\infty}_c(\mathbb{R}^{2d})$ such that $h|_{(1-\epsilon)Q}\equiv 1$ and $\m where $\mathcal{F}_W(R)=\frac{h}{|\Lambda|\cdot A(g,g)}$.

Theorems & Definitions (43)

  • Theorem 1.1: Proposition 2, GrPa13, Theorem 7.4, Sk20
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1: Theorem 2, Ca68
  • Example 2.2
  • Theorem 3.1: Proposition 2, GrPa13, Theorem 7.4, Sk20
  • proof
  • Definition 3.2
  • Proposition 3.3: Theorem 6.5, LaMc24
  • ...and 33 more