Modulation spaces: from Feichtinger's original definition to modern Gabor and symplectic characterizations

Abstract

We present an expository excursus on modulation spaces, from Feichtinger's original construction on locally compact abelian groups to recent symplectic formulations. We trace the same local-global principle through bounded uniform partitions of unity, the short-time Fourier transform, coorbit methods, and Gabor coefficients, while distinguishing equivalent descriptions from the weighted, quasi-Banach, and Gelfand-Shilov extensions of the underlying functional setting. We also discuss Gabor matrices, convolution and embedding estimates, and an application to a nonlinear dispersive equation. The final part is devoted to metaplectic Wigner distributions W_𝒜. In the shift-invertible case these representations are, up to chirps and a linear change of phase-space variables, rescaled short-time Fourier transforms. This structural fact explains both their characterization of modulation and Wiener amalgam spaces and their discretization by metaplectic Gabor frames, and places the recent symplectic theory naturally within Feichtinger's modulation-space framework.

Characterizations in terms of Generalized Inverses
Feroz Siddique
arXiv14 Oct 2026PreprintCC BY 4.0Full textAlgebra & number theoryarXiv
Characterizations of Toeplitz operators on the annulus
Shubham Jain, Jaydeb Sarkar
arXiv29 Oct 2026PreprintarXiv licenceFull textAnalysis & PDEsarXiv
Exact L² convergence through the wavelet and tent space characterizations of h^p(ℝⁿ), 0<p≤ 1
Mohamed El Aoumari
arXiv1 Oct 2026PreprintarXiv licenceFull textAnalysis & PDEsarXiv
Characterizations of φ-inner uniform domains and φ-uniform domains in metric spaces
Shu-Jing Gao, Miao He
arXiv29 Oct 2026PreprintarXiv licenceFull textAnalysis & PDEsarXiv
Characteristic-free characterizations of projective space for smooth toric varieties
Osamu Fujino, Hiroshi Sato
arXiv12 Sep 2026PreprintarXiv licenceFull textAlgebra & number theoryarXiv

Cite

BibTeX RIS

BibTeX

@misc{caputo2026modulation,
  title = {{Modulation spaces: from Feichtinger's original definition to modern Gabor and symplectic characterizations}},
  author = {Antonio Caputo and Elena Cordero and Gianluca Giachi and Luigi Rodino},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.33894},
  eprint = {2609.33894},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.33894},
}

RIS

TY  - UNPB
TI  - Modulation spaces: from Feichtinger's original definition to modern Gabor and symplectic characterizations
AU  - Antonio Caputo
AU  - Elena Cordero
AU  - Gianluca Giachi
AU  - Luigi Rodino
PY  - 2026
DA  - 2026/09/27
PB  - arXiv
DO  - 10.48550/arxiv.2609.33894
UR  - https://arxiv.org/abs/2609.33894
AB  - We present an expository excursus on modulation spaces, from Feichtinger's original construction on locally compact abelian groups to recent symplectic formulations. We trace the same local-global principle through bounded uniform partitions of unity, the short-time Fourier transform, coorbit methods, and Gabor coefficients, while distinguishing equivalent descriptions from the weighted, quasi-Banach, and Gelfand-Shilov extensions of the underlying functional setting. We also discuss Gabor matrices, convolution and embedding estimates, and an application to a nonlinear dispersive equation. The final part is devoted to metaplectic Wigner distributions W_𝒜. In the shift-invertible case these representations are, up to chirps and a linear change of phase-space variables, rescaled short-time Fourier transforms. This structural fact explains both their characterization of modulation and Wiener amalgam spaces and their discretization by metaplectic Gabor frames, and places the recent symplectic theory naturally within Feichtinger's modulation-space framework.
ER  -

Abstract

Related

Cite