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Subdyadic time-frequency analysis: Gabor frames, modulation spaces, and Miyachi multipliers

Vicente Vergara

TL;DR

This work develops a subdyadic time–frequency framework adapted to dispersive phases $|\xi|^{\alpha}$, introducing an anisotropic phase‑space metric, subdyadic blocks, and a deformed Gabor lattice that yields a stable $L^{2}$ frame and a dispersive STFT. It defines dispersive modulation spaces $M^{p,q}_{\alpha,\beta}$ anchored to the subdyadic geometry, proves window and lattice independence, and establishes duality and inclusions, with a model boundedness result for Miyachi multipliers on $M^{2,2}_{\alpha,\beta}$. A two‑sided Miyachi condition enables almost diagonalization of such multipliers and their boundedness on the dispersive scale, linking oscillatory symbols to global function spaces. The paper then introduces a subdyadic Gabor wavefront set $WF^{G}_{\alpha}$, proving its microlocal invariance and ellipticity under smooth order‑zero pseudodifferential operators, thus providing a unified tool for global microlocal analysis and stable high‑frequency numerical schemes in dispersive settings. Collectively, these results bridge subdyadic harmonic analysis, global microlocal analysis, and computational approaches for dispersive PDEs, with precise quantitative control of localization and decay in phase space.

Abstract

We present a time-frequency framework adapted to dispersive phase functions via a subdyadic geometry in phase space. On top of this geometry we construct stable Gabor frames with quantitative control of overlap, almost orthogonality, and off-diagonal decay. Based on these frames we introduce modulation spaces consistent with the subdyadic scale and establish window and lattice independence, identifications in the Hilbertian case, duality, and natural inclusion relations. Within this setting we develop a theory for two-sided Miyachi multipliers, relying on discrete almost diagonalization and Wiener-Jaffard type results for well-localized matrices, and obtain boundedness on weighted modulation spaces. Finally, we define a Gabor-type wavefront set adapted to the subdyadic geometry and prove its invariance and ellipticity with respect to smooth order-zero pseudodifferential operators. Taken together, these results provide a unified tool both for global microlocal analysis and for the design of stable numerical schemes in high-frequency regimes.

Subdyadic time-frequency analysis: Gabor frames, modulation spaces, and Miyachi multipliers

TL;DR

This work develops a subdyadic time–frequency framework adapted to dispersive phases , introducing an anisotropic phase‑space metric, subdyadic blocks, and a deformed Gabor lattice that yields a stable frame and a dispersive STFT. It defines dispersive modulation spaces anchored to the subdyadic geometry, proves window and lattice independence, and establishes duality and inclusions, with a model boundedness result for Miyachi multipliers on . A two‑sided Miyachi condition enables almost diagonalization of such multipliers and their boundedness on the dispersive scale, linking oscillatory symbols to global function spaces. The paper then introduces a subdyadic Gabor wavefront set , proving its microlocal invariance and ellipticity under smooth order‑zero pseudodifferential operators, thus providing a unified tool for global microlocal analysis and stable high‑frequency numerical schemes in dispersive settings. Collectively, these results bridge subdyadic harmonic analysis, global microlocal analysis, and computational approaches for dispersive PDEs, with precise quantitative control of localization and decay in phase space.

Abstract

We present a time-frequency framework adapted to dispersive phase functions via a subdyadic geometry in phase space. On top of this geometry we construct stable Gabor frames with quantitative control of overlap, almost orthogonality, and off-diagonal decay. Based on these frames we introduce modulation spaces consistent with the subdyadic scale and establish window and lattice independence, identifications in the Hilbertian case, duality, and natural inclusion relations. Within this setting we develop a theory for two-sided Miyachi multipliers, relying on discrete almost diagonalization and Wiener-Jaffard type results for well-localized matrices, and obtain boundedness on weighted modulation spaces. Finally, we define a Gabor-type wavefront set adapted to the subdyadic geometry and prove its invariance and ellipticity with respect to smooth order-zero pseudodifferential operators. Taken together, these results provide a unified tool both for global microlocal analysis and for the design of stable numerical schemes in high-frequency regimes.

Paper Structure

This paper contains 25 sections, 27 theorems, 355 equations.

Table of Contents

  1. Introduction
  2. Conventions and notation.
  3. Subdyadic geometry in phase space
  4. Natural scale for dispersive phases
  5. Subdyadic metric on the cotangent bundle
  6. Subdyadic blocks in phase space
  7. Subdyadic Gabor lattice and dispersive STFT
  8. Construction of a subdyadic lattice in phase space
  9. Dispersive wave packets and adapted Gabor transform
  10. Dispersive modulation spaces $M^{p,q}_{\alpha,\beta}$
  11. Low frequencies.
  12. Discrete definition via Gabor coefficients
  13. Independence of the window and the lattice
  14. Existence of $\alpha$–admissible pairs and norm equivalence
  15. Relation with $L^2$ and density of $\mathcal{S}$
  16. ...and 10 more sections

Key Result

Theorem 1.1

If $m:\mathbb{R}^{d}\setminus\{0\}\to\mathbb{C}$ satisfies Cond:Miya:2-sided for every $\gamma\in \mathbb{N}^d$ with $|\gamma|\leq N_d$, then where and

Theorems & Definitions (61)

  • Theorem 1.1: Corollary 11,BeltranBennett2017
  • Proposition 2.1: Anisotropic distance induced by $g^\alpha$
  • proof
  • Lemma 2.2: Anisotropic volume and doubling property
  • proof
  • Lemma 2.3: Covering with finite overlap
  • proof
  • Lemma 3.1: Frequency localization of the packets
  • proof
  • Definition 3.2: Separation and relative separation
  • ...and 51 more