More Uncertainty Principles for Metaplectic Time-Frequency Representations

TL;DR

The paper addresses how to transfer classical uncertainty principles from the short-time Fourier transform to metaplectic time-frequency representations $W_{m{ m A}}$, deriving Beurling-type, Hardy-type, and Gelfand–Shilov results. It develops a general strategy based on factorizations of symplectic matrices (notably the pre-Iwasawa decomposition) to reduce statements about $W_{m{ m A}}$ to partial STFT problems, and identifies a crucial dichotomy governed by whether the associated unitary block $U^tU$ is block-diagonal. The contributions include explicit Beurling-type and Hardy-type principles for $W_{m{ m A}}$, extensions to quadratic metaplectic representations and pairs of metaplectic operators, and quantitative forms such as Nazarov-type inequalities, along with relaxations and matrix-choice considerations. This framework provides a versatile, scalable method for establishing uncertainty principles across a broad class of metaplectic time-frequency representations with potential implications for pseudodifferential operator theory and function space analysis.

Abstract

We develop a method for the transfer of an uncertainty principle for the short-time Fourier transform or a Fourier pair to an uncertainty principle for a sesquilinear or quadratic metaplectic time-frequency representation. In particular, we derive Beurling-type and Hardy-type uncertainty principles for metaplectic time-frequency representations.

Theorems & Definitions (34)

• Proposition 2.1: Pre-Iwasawa decomposition
• Definition 3.1: Partial short-time Fourier transform
• Lemma 3.2
• Theorem 3.3
• Theorem 3.4
• proof
• Theorem 3.5
• proof
• Theorem 3.6
• proof