Publications
Preprints
2.
Getter, Max; Trapasso, S. Ivan
The stability landscape in wave-packet scattering: geometric rigidity and sharp Sobolev thresholds Preprint
2026.
Abstract | Links | BibTeX | Tags: deformation stability, harmonic analysis, machine learning, microlocal analysis, multiscale analysis, scattering transform
@workingpaper{GT_2026,
title = {The stability landscape in wave-packet scattering: geometric rigidity and sharp Sobolev thresholds},
author = {Max Getter and S. Ivan Trapasso},
url = {https://arxiv.org/abs/2607.21578, arXiv},
year = {2026},
date = {2026-07-24},
urldate = {2026-07-24},
abstract = {A central challenge in modern harmonic analysis is to quantify the balance between the approximation power of finely resolved multiscale representations and their robustness to nonlinear changes of coordinates, a problem arising naturally in signal processing and partial differential equations. Motivated by Mallat's pioneering results on the wavelet scattering transform, we identify a sharp resolution--robustness trade-off for scattering-type nonlinear multiscale representations built upon general wave-packet systems, showing that stability under small diffeomorphisms is governed by the geometry of the underlying frequency decomposition. In particular, for wave-packet systems with finer transverse resolution than wavelets, including curvelets and shearlets, we establish a geometric rigidity phenomenon: arbitrarily small, smooth, compactly supported deformations can move high-frequency mass across adjacent channels, leading to instability already at the first scattering layer. We complement this obstruction by identifying the sharp Sobolev threshold for deformation stability: below the critical regularity no Mallat-type estimate can hold, while at and above it stability is recovered by means of matched commutator bounds that allow deformations to be propagated through the frequency channels. Together, these results provide a systematic deformation-stability theory for Euclidean scattering transforms and yield the first stability estimates intrinsic to the scattering architecture beyond the classical wavelet setting.},
keywords = {deformation stability, harmonic analysis, machine learning, microlocal analysis, multiscale analysis, scattering transform},
pubstate = {published},
tppubtype = {workingpaper}
}
A central challenge in modern harmonic analysis is to quantify the balance between the approximation power of finely resolved multiscale representations and their robustness to nonlinear changes of coordinates, a problem arising naturally in signal processing and partial differential equations. Motivated by Mallat's pioneering results on the wavelet scattering transform, we identify a sharp resolution–robustness trade-off for scattering-type nonlinear multiscale representations built upon general wave-packet systems, showing that stability under small diffeomorphisms is governed by the geometry of the underlying frequency decomposition. In particular, for wave-packet systems with finer transverse resolution than wavelets, including curvelets and shearlets, we establish a geometric rigidity phenomenon: arbitrarily small, smooth, compactly supported deformations can move high-frequency mass across adjacent channels, leading to instability already at the first scattering layer. We complement this obstruction by identifying the sharp Sobolev threshold for deformation stability: below the critical regularity no Mallat-type estimate can hold, while at and above it stability is recovered by means of matched commutator bounds that allow deformations to be propagated through the frequency channels. Together, these results provide a systematic deformation-stability theory for Euclidean scattering transforms and yield the first stability estimates intrinsic to the scattering architecture beyond the classical wavelet setting.
1.
Alberti, Giovanni S.; Felisi, Alessandro; Santacesaria, Matteo; Trapasso, S. Ivan
Compressed sensing for inverse problems II: applications to deconvolution, source recovery, and MRI Preprint
2025.
Abstract | Links | BibTeX | Tags: compressed sensing, harmonic analysis, inverse problems, mathematical imaging, sparse recovery
@workingpaper{AFST_CSIP2_25,
title = {Compressed sensing for inverse problems II: applications to deconvolution, source recovery, and MRI},
author = {Giovanni S. Alberti and Alessandro Felisi and Matteo Santacesaria and S. Ivan Trapasso},
url = {https://arxiv.org/abs/2501.01929, arXiv},
year = {2025},
date = {2025-01-03},
urldate = {2025-01-03},
abstract = {This paper extends the sample complexity theory for ill-posed inverse problems developed in a recent work by the authors [`Compressed sensing for inverse problems and the sample complexity of the sparse Radon transform', J. Eur. Math. Soc., to appear], which was originally focused on the sparse Radon transform. We demonstrate that the underlying abstract framework, based on infinite-dimensional compressed sensing and generalized sampling techniques, can effectively handle a variety of practical applications. Specifically, we analyze three case studies: (1) The reconstruction of a sparse signal from a finite number of pointwise blurred samples; (2) The recovery of the (sparse) source term of an elliptic partial differential equation from finite samples of the solution; and (3) A moderately ill-posed variation of the classical sensing problem of recovering a wavelet-sparse signal from finite Fourier samples, motivated by magnetic resonance imaging. For each application, we establish rigorous recovery guarantees by verifying the key theoretical requirements, including quasi-diagonalization and coherence bounds. Our analysis reveals that careful consideration of balancing properties and optimized sampling strategies can lead to improved reconstruction performance. The results provide a unified theoretical foundation for compressed sensing approaches to inverse problems while yielding practical insights for specific applications.},
keywords = {compressed sensing, harmonic analysis, inverse problems, mathematical imaging, sparse recovery},
pubstate = {published},
tppubtype = {workingpaper}
}
This paper extends the sample complexity theory for ill-posed inverse problems developed in a recent work by the authors [`Compressed sensing for inverse problems and the sample complexity of the sparse Radon transform’, J. Eur. Math. Soc., to appear], which was originally focused on the sparse Radon transform. We demonstrate that the underlying abstract framework, based on infinite-dimensional compressed sensing and generalized sampling techniques, can effectively handle a variety of practical applications. Specifically, we analyze three case studies: (1) The reconstruction of a sparse signal from a finite number of pointwise blurred samples; (2) The recovery of the (sparse) source term of an elliptic partial differential equation from finite samples of the solution; and (3) A moderately ill-posed variation of the classical sensing problem of recovering a wavelet-sparse signal from finite Fourier samples, motivated by magnetic resonance imaging. For each application, we establish rigorous recovery guarantees by verifying the key theoretical requirements, including quasi-diagonalization and coherence bounds. Our analysis reveals that careful consideration of balancing properties and optimized sampling strategies can lead to improved reconstruction performance. The results provide a unified theoretical foundation for compressed sensing approaches to inverse problems while yielding practical insights for specific applications.
Journal Articles
11.
Mazzucchi, Sonia; Nicola, Fabio; Trapasso, S. Ivan
Phase space analysis of finite and infinite dimensional Fresnel integrals Journal Article
In: J. Funct. Anal., vol. 289, iss. 8, no. 111009, pp. 1–51, 2025.
Abstract | Links | BibTeX | Tags: Feynman path integrals, harmonic analysis, infinite dimensional analysis, mathematical physics, modulation spaces
@article{MNT_JFA_25,
title = {Phase space analysis of finite and infinite dimensional Fresnel integrals},
author = {Sonia Mazzucchi and Fabio Nicola and S. Ivan Trapasso},
url = {https://arxiv.org/abs/2403.20082, arXiv
https://www.sciencedirect.com/science/article/pii/S0022123625001910/pdfft?md5=2323ac7b3fed88926bde9dafa29aba1c&pid=1-s2.0-S0022123625001910-main.pdf, PDF},
doi = {10.1016/j.jfa.2025.111009},
year = {2025},
date = {2025-04-15},
urldate = {2025-04-15},
journal = {J. Funct. Anal.},
volume = {289},
number = {111009},
issue = {8},
pages = {1–51},
abstract = {The full characterization of the class of Fresnel integrable functions is an open problem in functional analysis, with significant applications to mathematical physics (Feynman path integrals) and the analysis of the Schrödinger equation. In finite dimension, we prove the Fresnel integrability of functions in the Sjöstrand class (M^{infty,1}) - a family of continuous and bounded functions, locally enjoying the mild regularity of the Fourier transform of an integrable function. This result broadly extends the current knowledge on the Fresnel integrability of Fourier transforms of finite complex measures, and relies upon ideas and techniques of Gabor wave packet analysis. We also discuss the problem of designing infinite-dimensional extensions of this result, obtaining the first, non-trivial concrete realization of a general framework of projective functional extensions introduced by Albeverio and Mazzucchi. As an interesting byproduct, we obtain the exact (M^{infty,1} to L^infty) operator norm of the free Schrödinger evolution operator.},
keywords = {Feynman path integrals, harmonic analysis, infinite dimensional analysis, mathematical physics, modulation spaces},
pubstate = {published},
tppubtype = {article}
}
The full characterization of the class of Fresnel integrable functions is an open problem in functional analysis, with significant applications to mathematical physics (Feynman path integrals) and the analysis of the Schrödinger equation. In finite dimension, we prove the Fresnel integrability of functions in the Sjöstrand class \(M^{\infty,1}\) – a family of continuous and bounded functions, locally enjoying the mild regularity of the Fourier transform of an integrable function. This result broadly extends the current knowledge on the Fresnel integrability of Fourier transforms of finite complex measures, and relies upon ideas and techniques of Gabor wave packet analysis. We also discuss the problem of designing infinite-dimensional extensions of this result, obtaining the first, non-trivial concrete realization of a general framework of projective functional extensions introduced by Albeverio and Mazzucchi. As an interesting byproduct, we obtain the exact \(M^{\infty,1} \to L^\infty\) operator norm of the free Schrödinger evolution operator.
10.
Nicola, Fabio; Trapasso, S. Ivan
Generalized moduli of continuity under irregular or random deformations via multiscale analysis Journal Article
In: Inf. Inference, vol. 14, iss. 2, no. iaaf006, 2025.
Abstract | Links | BibTeX | Tags: deformation stability, harmonic analysis, machine learning, multiscale analysis
@article{NT_IMAIAI_25,
title = {Generalized moduli of continuity under irregular or random deformations via multiscale analysis},
author = {Fabio Nicola and S. Ivan Trapasso},
url = {https://arxiv.org/abs/2104.11977, arXiv
https://academic.oup.com/imaiai/article/doi/10.1093/imaiai/iaaf006/8108050?utm_source=authortollfreelink&utm_campaign=imaiai&utm_medium=email&guestAccessKey=87ddedf1-38c9-401d-b53d-cf41144903cf, PDF},
doi = {10.1093/imaiai/iaaf006},
year = {2025},
date = {2025-04-08},
urldate = {2025-04-08},
journal = {Inf. Inference},
volume = {14},
number = {iaaf006},
issue = {2},
abstract = {Motivated by the problem of robustness to deformations of the input for deep convolutional neural networks, we identify signal classes which are inherently stable to irregular deformations induced by distortion fields (tauin L^infty(mathbb{R}^d;mathbb{R}^d)), to be characterized in terms of a generalized modulus of continuity associated with the deformation operator. Resorting to ideas of harmonic and multiscale analysis, we prove that for signals in multiresolution approximation spaces (U_s) at scale (s), stability in (L^2) holds in the regime (|tau|_{L^infty}/sll 1) - essentially as an effect of the uncertainty principle. Instability occurs when (|tau|_{L^infty}/sgg 1), and we provide a sharp upper bound for the asymptotic growth rate. The stability results are then extended to signals in the Besov space (B^{d/2}_{2,1}) tailored to the given multiresolution approximation. We also consider the case of more general time-frequency deformations. Finally, we provide stochastic versions of the aforementioned results, namely we study the issue of stability in mean when (tau(x)) is modeled as a random field (not bounded, in general) with identically distributed variables (|tau(x)|), (xinmathbb{R}^d).},
keywords = {deformation stability, harmonic analysis, machine learning, multiscale analysis},
pubstate = {published},
tppubtype = {article}
}
Motivated by the problem of robustness to deformations of the input for deep convolutional neural networks, we identify signal classes which are inherently stable to irregular deformations induced by distortion fields \(\tau\in L^\infty(\mathbb{R}^d;\mathbb{R}^d)\), to be characterized in terms of a generalized modulus of continuity associated with the deformation operator. Resorting to ideas of harmonic and multiscale analysis, we prove that for signals in multiresolution approximation spaces \(U_s\) at scale \(s\), stability in \(L^2\) holds in the regime \(\|\tau\|_{L^\infty}/s\ll 1\) – essentially as an effect of the uncertainty principle. Instability occurs when \(\|\tau\|_{L^\infty}/s\gg 1\), and we provide a sharp upper bound for the asymptotic growth rate. The stability results are then extended to signals in the Besov space \(B^{d/2}_{2,1}\) tailored to the given multiresolution approximation. We also consider the case of more general time-frequency deformations. Finally, we provide stochastic versions of the aforementioned results, namely we study the issue of stability in mean when \(\tau(x)\) is modeled as a random field (not bounded, in general) with identically distributed variables \(|\tau(x)|\), \(x\in\mathbb{R}^d\).
9.
Alberti, Giovanni S.; Felisi, Alessandro; Santacesaria, Matteo; Trapasso, S. Ivan
Compressed sensing for inverse problems and the sample complexity of the sparse Radon transform Journal Article
In: J. Eur. Math. Soc. (JEMS), iss. to appear, 2025.
Abstract | Links | BibTeX | Tags: compressed sensing, harmonic analysis, inverse problems, mathematical imaging, Radon transform, sparse recovery
@article{AFST_JEMS_24,
title = {Compressed sensing for inverse problems and the sample complexity of the sparse Radon transform},
author = {Giovanni S. Alberti and Alessandro Felisi and Matteo Santacesaria and S. Ivan Trapasso},
url = {https://arxiv.org/abs/2302.03577, arXiv},
doi = {10.4171/jems/1617},
year = {2025},
date = {2025-04-01},
urldate = {2025-04-01},
journal = {J. Eur. Math. Soc. (JEMS)},
issue = {to appear},
abstract = {Compressed sensing allows for the recovery of sparse signals from few measurements, whose number is proportional to the sparsity of the unknown signal, up to logarithmic factors. The classical theory typically considers either random linear measurements or subsampled isometries and has found many applications, including accelerated magnetic resonance imaging, which is modeled by the subsampled Fourier transform. In this work, we develop a general theory of infinite-dimensional compressed sensing for abstract inverse problems, possibly ill-posed, involving an arbitrary forward operator. This is achieved by considering a generalized restricted isometry property, and a quasi-diagonalization property of the forward map.
As a notable application, for the first time, we obtain rigorous recovery estimates for the sparse Radon transform (i.e., with a finite number of angles (theta_1,dots,theta_m)), which models computed tomography, in both the parallel-beam and the fan-beam settings. In the case when the unknown signal is (s)-sparse with respect to an orthonormal basis of compactly supported wavelets, we prove stable recovery under the condition [ mgtrsimß, ] up to logarithmic factors.},
keywords = {compressed sensing, harmonic analysis, inverse problems, mathematical imaging, Radon transform, sparse recovery},
pubstate = {published},
tppubtype = {article}
}
Compressed sensing allows for the recovery of sparse signals from few measurements, whose number is proportional to the sparsity of the unknown signal, up to logarithmic factors. The classical theory typically considers either random linear measurements or subsampled isometries and has found many applications, including accelerated magnetic resonance imaging, which is modeled by the subsampled Fourier transform. In this work, we develop a general theory of infinite-dimensional compressed sensing for abstract inverse problems, possibly ill-posed, involving an arbitrary forward operator. This is achieved by considering a generalized restricted isometry property, and a quasi-diagonalization property of the forward map.
As a notable application, for the first time, we obtain rigorous recovery estimates for the sparse Radon transform (i.e., with a finite number of angles \(\theta_1,\dots,\theta_m\)), which models computed tomography, in both the parallel-beam and the fan-beam settings. In the case when the unknown signal is \(s\)-sparse with respect to an orthonormal basis of compactly supported wavelets, we prove stable recovery under the condition \[ m\gtrsimß, \] up to logarithmic factors.
As a notable application, for the first time, we obtain rigorous recovery estimates for the sparse Radon transform (i.e., with a finite number of angles \(\theta_1,\dots,\theta_m\)), which models computed tomography, in both the parallel-beam and the fan-beam settings. In the case when the unknown signal is \(s\)-sparse with respect to an orthonormal basis of compactly supported wavelets, we prove stable recovery under the condition \[ m\gtrsimß, \] up to logarithmic factors.
8.
Trapasso, S. Ivan
Phase space analysis of spectral multipliers for the twisted Laplacian Journal Article
In: Trans. Amer. Math. Soc., vol. 378, no. 2, pp. 967-999, 2025.
Abstract | Links | BibTeX | Tags: harmonic analysis, harmonic oscillator, modulation spaces, partial differential equations, spectral multipliers
@article{T_TAMS_24,
title = {Phase space analysis of spectral multipliers for the twisted Laplacian},
author = {S. Ivan Trapasso},
url = {https://arxiv.org/abs/2306.00592, arXiv},
doi = {10.1090/tran/9224},
year = {2025},
date = {2025-02-01},
urldate = {2025-02-01},
journal = {Trans. Amer. Math. Soc.},
volume = {378},
number = {2},
pages = {967-999},
abstract = {We prove boundedness results on modulation and Wiener amalgam spaces for some families of spectral multipliers for the twisted Laplacian. We exploit the metaplectic equivalence relating the twisted Laplacian with a partial harmonic oscillator, leading to a general transference principle for the corresponding spectral multipliers. Our analysis encompasses powers of the twisted Laplacian and oscillating multipliers, with applications to the corresponding Schrödinger and wave flows. On the other hand, elaborating on the twisted convolution structure of the eigenprojections and its connection with the Weyl product of symbols, we obtain a complete picture of the boundedness of the heat flow for the twisted Laplacian. Results of the same kind are established for fractional heat flows via subordination.},
keywords = {harmonic analysis, harmonic oscillator, modulation spaces, partial differential equations, spectral multipliers},
pubstate = {published},
tppubtype = {article}
}
We prove boundedness results on modulation and Wiener amalgam spaces for some families of spectral multipliers for the twisted Laplacian. We exploit the metaplectic equivalence relating the twisted Laplacian with a partial harmonic oscillator, leading to a general transference principle for the corresponding spectral multipliers. Our analysis encompasses powers of the twisted Laplacian and oscillating multipliers, with applications to the corresponding Schrödinger and wave flows. On the other hand, elaborating on the twisted convolution structure of the eigenprojections and its connection with the Weyl product of symbols, we obtain a complete picture of the boundedness of the heat flow for the twisted Laplacian. Results of the same kind are established for fractional heat flows via subordination.
7.
Nicola, Fabio; Trapasso, S. Ivan
Stability of the scattering transform for deformations with minimal regularity Journal Article
In: J. Math. Pures Appl., vol. 180, pp. 122–150, 2023.
Abstract | Links | BibTeX | Tags: deformation stability, harmonic analysis, machine learning, multiscale analysis, scattering transform
@article{NT_JMPA_23,
title = {Stability of the scattering transform for deformations with minimal regularity},
author = {Fabio Nicola and S. Ivan Trapasso},
url = {https://arxiv.org/abs/2205.11142, arXiv
https://www.sciencedirect.com/science/article/pii/S0021782423001496/pdfft?md5=b95043909dc30fb3845bb0cac8a65b05&pid=1-s2.0-S0021782423001496-main.pdf, PDF},
doi = {10.1016/j.matpur.2023.10.008},
year = {2023},
date = {2023-12-31},
urldate = {2023-12-31},
journal = {J. Math. Pures Appl.},
volume = {180},
pages = {122–150},
publisher = {Elsevier Masson},
abstract = {Within the mathematical analysis of deep convolutional neural networks, the wavelet scattering transform introduced by Stéphane Mallat is a unique example of how the ideas of multiscale analysis can be combined with a cascade of modulus nonlinearities to build a nonexpansive, translation invariant signal representation with provable geometric stability properties, namely Lipschitz continuity to the action of small (C^2) diffeomorphisms - a remarkable result for both theoretical and practical purposes, inherently depending on the choice of the filters and their arrangement into a hierarchical architecture. In this note, we further investigate the intimate relationship between the scattering structure and the regularity of the deformation in the Hölder regularity scale (C^alpha), (alpha >0). We are able to precisely identify the stability threshold, proving that stability is still achievable for deformations of class (C^{alpha}), (alpha>1), whereas instability phenomena can occur at lower regularity levels modelled by (C^alpha), (0le alpha <1). While the behaviour at the threshold given by Lipschitz (or even (C^1)) regularity remains beyond reach, we are able to prove a stability bound in that case, up to (varepsilon) losses.},
keywords = {deformation stability, harmonic analysis, machine learning, multiscale analysis, scattering transform},
pubstate = {published},
tppubtype = {article}
}
Within the mathematical analysis of deep convolutional neural networks, the wavelet scattering transform introduced by Stéphane Mallat is a unique example of how the ideas of multiscale analysis can be combined with a cascade of modulus nonlinearities to build a nonexpansive, translation invariant signal representation with provable geometric stability properties, namely Lipschitz continuity to the action of small \(C^2\) diffeomorphisms – a remarkable result for both theoretical and practical purposes, inherently depending on the choice of the filters and their arrangement into a hierarchical architecture. In this note, we further investigate the intimate relationship between the scattering structure and the regularity of the deformation in the Hölder regularity scale \(C^\alpha\), \(\alpha >0\). We are able to precisely identify the stability threshold, proving that stability is still achievable for deformations of class \(C^{\alpha}\), \(\alpha>1\), whereas instability phenomena can occur at lower regularity levels modelled by \(C^\alpha\), \(0\le \alpha <1\). While the behaviour at the threshold given by Lipschitz (or even \(C^1\)) regularity remains beyond reach, we are able to prove a stability bound in that case, up to \(\varepsilon\) losses.
6.
Nicola, Fabio; Romero, José Luis; Trapasso, S. Ivan
On the existence of optimizers for time–frequency concentration problems Journal Article
In: Calc. Var. Partial Differential Equations, vol. 62, no. 1, pp. 1–21, 2023.
Abstract | Links | BibTeX | Tags: concentration problems, harmonic analysis, time-frequency analysis, time-frequency distributions
@article{NRT_CVPDE_23,
title = {On the existence of optimizers for time–frequency concentration problems},
author = {Fabio Nicola and José Luis Romero and S. Ivan Trapasso},
url = {https://arxiv.org/abs/2112.09675, arXiv
https://rdcu.be/cZmhU, PDF},
doi = {10.1007/s00526-022-02358-6},
year = {2023},
date = {2023-08-31},
urldate = {2023-08-31},
journal = {Calc. Var. Partial Differential Equations},
volume = {62},
number = {1},
pages = {1–21},
publisher = {Springer Berlin Heidelberg},
abstract = {We consider the problem of the maximum concentration in a fixed measurable subset (Omegasubsetmathbb{R}^{2d}) of the time-frequency space for functions (fin L^2(mathbb{R}^{d})). The notion of concentration can be made mathematically precise by considering the (L^p)-norm on (Omega) of some time-frequency distribution of (f) such as the ambiguity function (A(f)). We provide a positive answer to an open maximization problem, by showing that for every subset (Omegasubsetmathbb{R}^{2d}) of finite measure and every (1leq p<infty), there exists an optimizer for [
sup{|A(f)|_{L^p(Omega)}: fin L^2(mathbb{R}^{d}), |f|_{L^2}=1
}.
] The lack of weak upper semicontinuity and the invariance under time-frequency shifts make the problem challenging. The proof is based on concentration compactness with time-frequency shifts as dislocations, and certain integral bounds and asymptotic decoupling estimates for the ambiguity function. We also discuss the case (p=infty) and related optimization problems for the time correlation function, the cross-ambiguity function with a fixed window, and for functions in the modulation spaces (M^q(mathbb{R}^{d})), (0<q<2), equipped with continuous or discrete-type (quasi-)norms.},
keywords = {concentration problems, harmonic analysis, time-frequency analysis, time-frequency distributions},
pubstate = {published},
tppubtype = {article}
}
We consider the problem of the maximum concentration in a fixed measurable subset \(\Omega\subset\mathbb{R}^{2d}\) of the time-frequency space for functions \(f\in L^2(\mathbb{R}^{d})\). The notion of concentration can be made mathematically precise by considering the \(L^p\)-norm on \(\Omega\) of some time-frequency distribution of \(f\) such as the ambiguity function \(A(f)\). We provide a positive answer to an open maximization problem, by showing that for every subset \(\Omega\subset\mathbb{R}^{2d}\) of finite measure and every \(1\leq p<\infty\), there exists an optimizer for \[
\sup\{\|A(f)\|_{L^p(\Omega)}:\ f\in L^2(\mathbb{R}^{d}),\ \|f\|_{L^2}=1
\}.
\] The lack of weak upper semicontinuity and the invariance under time-frequency shifts make the problem challenging. The proof is based on concentration compactness with time-frequency shifts as dislocations, and certain integral bounds and asymptotic decoupling estimates for the ambiguity function. We also discuss the case \(p=\infty\) and related optimization problems for the time correlation function, the cross-ambiguity function with a fixed window, and for functions in the modulation spaces \(M^q(\mathbb{R}^{d})\), \(0<q<2\), equipped with continuous or discrete-type (quasi-)norms.
\sup\{\|A(f)\|_{L^p(\Omega)}:\ f\in L^2(\mathbb{R}^{d}),\ \|f\|_{L^2}=1
\}.
\] The lack of weak upper semicontinuity and the invariance under time-frequency shifts make the problem challenging. The proof is based on concentration compactness with time-frequency shifts as dislocations, and certain integral bounds and asymptotic decoupling estimates for the ambiguity function. We also discuss the case \(p=\infty\) and related optimization problems for the time correlation function, the cross-ambiguity function with a fixed window, and for functions in the modulation spaces \(M^q(\mathbb{R}^{d})\), \(0<q<2\), equipped with continuous or discrete-type (quasi-)norms.
5.
Nicola, Fabio; Trapasso, S. Ivan
A note on the HRT conjecture and a new uncertainty principle for the short-time Fourier transform Journal Article
In: J. Fourier Anal. Appl., vol. 26, no. 4, pp. 68, 2020.
Abstract | Links | BibTeX | Tags: Gabor analysis, harmonic analysis, time-frequency analysis
@article{NT_JFAA_20,
title = {A note on the HRT conjecture and a new uncertainty principle for the short-time Fourier transform},
author = {Fabio Nicola and S. Ivan Trapasso},
url = {https://arxiv.org/abs/1911.12241, arXiv},
doi = {10.1007/s00041-020-09769-z},
year = {2020},
date = {2020-07-30},
urldate = {2020-07-30},
journal = {J. Fourier Anal. Appl.},
volume = {26},
number = {4},
pages = {68},
publisher = {Springer US New York},
abstract = {In this note we provide a negative answer to a question raised by M. Kreisel concerning a condition on the short-time Fourier transform that would imply the HRT conjecture. In particular we provide a new type of uncertainty principle for the short-time Fourier transform which forbids the arrangement of an arbitrary "bump with fat tail" profile.},
keywords = {Gabor analysis, harmonic analysis, time-frequency analysis},
pubstate = {published},
tppubtype = {article}
}
In this note we provide a negative answer to a question raised by M. Kreisel concerning a condition on the short-time Fourier transform that would imply the HRT conjecture. In particular we provide a new type of uncertainty principle for the short-time Fourier transform which forbids the arrangement of an arbitrary “bump with fat tail” profile.
4.
Cordero, Elena; Trapasso, S. Ivan
Linear perturbations of the Wigner distribution and the Cohen class Journal Article
In: Anal. Appl., vol. 18, no. 03, pp. 385–422, 2020.
Abstract | Links | BibTeX | Tags: harmonic analysis, time-frequency analysis, time-frequency distributions, Wigner distributions
@article{CT_AA_20,
title = {Linear perturbations of the Wigner distribution and the Cohen class},
author = {Elena Cordero and S. Ivan Trapasso},
url = {https://arxiv.org/abs/1811.07795, arXiv},
doi = {10.1142/S0219530519500052},
year = {2020},
date = {2020-05-01},
urldate = {2020-05-01},
journal = {Anal. Appl.},
volume = {18},
number = {03},
pages = {385–422},
publisher = {World Scientific Publishing Company},
abstract = {The Wigner distribution is a milestone of Time-frequency Analysis. In order to cope with its drawbacks while preserving the desirable features that made it so popular, several kind of modifications have been proposed. This contributions fits into this perspective. We introduce a family of phase-space representations of Wigner type associated with invertible matrices and explore their general properties. As main result, we provide a characterization for the Cohen's class. This feature suggests to interpret this family of representations as linear perturbations of the Wigner distribution. We show which of its properties survive under linear perturbations and which ones are truly distinctive of its central role.},
keywords = {harmonic analysis, time-frequency analysis, time-frequency distributions, Wigner distributions},
pubstate = {published},
tppubtype = {article}
}
The Wigner distribution is a milestone of Time-frequency Analysis. In order to cope with its drawbacks while preserving the desirable features that made it so popular, several kind of modifications have been proposed. This contributions fits into this perspective. We introduce a family of phase-space representations of Wigner type associated with invertible matrices and explore their general properties. As main result, we provide a characterization for the Cohen’s class. This feature suggests to interpret this family of representations as linear perturbations of the Wigner distribution. We show which of its properties survive under linear perturbations and which ones are truly distinctive of its central role.
3.
Cordero, Elena; Nicola, Fabio; Trapasso, S. Ivan
Almost diagonalization of τ-pseudodifferential operators with symbols in Wiener amalgam and modulation spaces Journal Article
In: J. Fourier Anal. Appl., vol. 25, no. 4, pp. 1927–1957, 2019.
Abstract | Links | BibTeX | Tags: almost diagonalization, harmonic analysis, modulation spaces, pseudodifferential operators, time-frequency analysis
@article{CNT_JFAA_19,
title = {Almost diagonalization of τ-pseudodifferential operators with symbols in Wiener amalgam and modulation spaces},
author = {Elena Cordero and Fabio Nicola and S. Ivan Trapasso},
url = {https://arxiv.org/abs/1802.10314, arXiv},
doi = {10.1007/s00041-018-09651-z},
year = {2019},
date = {2019-08-15},
urldate = {2019-08-15},
journal = {J. Fourier Anal. Appl.},
volume = {25},
number = {4},
pages = {1927–1957},
publisher = {Springer US},
abstract = {In this paper we focus on the almost-diagonalization properties of (tau)-pseudodifferential operators using techniques from time-frequency analysis. Our function spaces are modulation spaces and the special class of Wiener amalgam spaces arising by considering the action of the Fourier transform of modulation spaces. A particular example is provided by the Sjöstrand class, for which Gröchenig exhibited the almost diagonalization of Weyl operators. We shall show that such result can be extended to any (tau)-pseudodifferential operator, for (tau in [0,1]), also with symbol in weighted Wiener amalgam spaces. As a consequence, we infer boundedness, algebra and Wiener properties for (tau)-pseudodifferential operators on Wiener amalgam and modulation spaces.},
keywords = {almost diagonalization, harmonic analysis, modulation spaces, pseudodifferential operators, time-frequency analysis},
pubstate = {published},
tppubtype = {article}
}
In this paper we focus on the almost-diagonalization properties of \(\tau\)-pseudodifferential operators using techniques from time-frequency analysis. Our function spaces are modulation spaces and the special class of Wiener amalgam spaces arising by considering the action of the Fourier transform of modulation spaces. A particular example is provided by the Sjöstrand class, for which Gröchenig exhibited the almost diagonalization of Weyl operators. We shall show that such result can be extended to any \(\tau\)-pseudodifferential operator, for \(\tau \in [0,1]\), also with symbol in weighted Wiener amalgam spaces. As a consequence, we infer boundedness, algebra and Wiener properties for \(\tau\)-pseudodifferential operators on Wiener amalgam and modulation spaces.
2.
Cordero, Elena; D'Elia, Lorenza; Trapasso, S. Ivan
Norm estimates for τ-pseudodifferential operators in Wiener amalgam and modulation spaces Journal Article
In: J. Math. Anal. Appl., vol. 471, no. 1-2, pp. 541–563, 2019.
Abstract | Links | BibTeX | Tags: harmonic analysis, modulation spaces, pseudodifferential operators, time-frequency analysis, Wiener amalgam spaces
@article{CDT_JMAA_19,
title = {Norm estimates for τ-pseudodifferential operators in Wiener amalgam and modulation spaces},
author = {Elena Cordero and Lorenza D'Elia and S. Ivan Trapasso},
url = {https://arxiv.org/abs/1803.07865, arXiv
https://www.sciencedirect.com/science/article/pii/S0022247X18309193/pdfft?md5=56e69bc9745c316f260d37420041913d&pid=1-s2.0-S0022247X18309193-main.pdf, PDF},
doi = {10.1016/j.jmaa.2018.10.090},
year = {2019},
date = {2019-03-01},
urldate = {2019-03-01},
journal = {J. Math. Anal. Appl.},
volume = {471},
number = {1-2},
pages = {541–563},
publisher = {Academic Press},
abstract = {We study continuity properties on modulation spaces for (tau)-pseudodifferential operators with symbols (a) in Wiener amalgam spaces. We obtain boundedness results for (tau in (0,1)) whereas, in the end-points (tau=0) and (tau=1), the corresponding operators are in general unbounded. Furthermore, for (tau in (0,1)), we exhibit a function of (tau) which is an upper bound for the operator norm.
The continuity properties of (tau)-pseudodifferential operators, for any (tauin [0,1]), with symbols (a) in modulation spaces are well known. Here we find an upper bound for the operator norm which does not depend on the parameter (tau in [0,1]), as expected. Key ingredients are uniform continuity estimates for (tau)-Wigner distributions.},
keywords = {harmonic analysis, modulation spaces, pseudodifferential operators, time-frequency analysis, Wiener amalgam spaces},
pubstate = {published},
tppubtype = {article}
}
We study continuity properties on modulation spaces for \(\tau\)-pseudodifferential operators with symbols \(a\) in Wiener amalgam spaces. We obtain boundedness results for \(\tau \in (0,1)\) whereas, in the end-points \(\tau=0\) and \(\tau=1\), the corresponding operators are in general unbounded. Furthermore, for \(\tau \in (0,1)\), we exhibit a function of \(\tau\) which is an upper bound for the operator norm.
The continuity properties of \(\tau\)-pseudodifferential operators, for any \(\tau\in [0,1]\), with symbols \(a\) in modulation spaces are well known. Here we find an upper bound for the operator norm which does not depend on the parameter \(\tau \in [0,1]\), as expected. Key ingredients are uniform continuity estimates for \(\tau\)-Wigner distributions.
The continuity properties of \(\tau\)-pseudodifferential operators, for any \(\tau\in [0,1]\), with symbols \(a\) in modulation spaces are well known. Here we find an upper bound for the operator norm which does not depend on the parameter \(\tau \in [0,1]\), as expected. Key ingredients are uniform continuity estimates for \(\tau\)-Wigner distributions.
1.
D'Elia, Lorenza; Trapasso, S. Ivan
Boundedness of pseudodifferential operators with symbols in Wiener amalgam spaces on modulation spaces Journal Article
In: J. Pseudo-Differ. Oper. Appl., vol. 9, pp. 881–890, 2018.
Abstract | Links | BibTeX | Tags: harmonic analysis, modulation spaces, pseudodifferential operators, time-frequency analysis, Wiener amalgam spaces
@article{DT_JPDO_18,
title = {Boundedness of pseudodifferential operators with symbols in Wiener amalgam spaces on modulation spaces},
author = {Lorenza D'Elia and S. Ivan Trapasso},
url = {https://arxiv.org/abs/1703.08989, arXiv},
doi = {10.1007/s11868-017-0220-1},
year = {2018},
date = {2018-12-01},
urldate = {2018-12-01},
journal = {J. Pseudo-Differ. Oper. Appl.},
volume = {9},
pages = {881–890},
publisher = {Springer International Publishing},
abstract = {This paper provides sufficient conditions for the boundedness of Weyl operators on modulation spaces. The Weyl symbols belong to Wiener amalgam spaces, or generalized modulation spaces, as recently renamed by their inventor Hans Feichtinger. This is the first result which relates symbols in Wiener amalgam spaces to operators acting on classical modulation spaces.},
keywords = {harmonic analysis, modulation spaces, pseudodifferential operators, time-frequency analysis, Wiener amalgam spaces},
pubstate = {published},
tppubtype = {article}
}
This paper provides sufficient conditions for the boundedness of Weyl operators on modulation spaces. The Weyl symbols belong to Wiener amalgam spaces, or generalized modulation spaces, as recently renamed by their inventor Hans Feichtinger. This is the first result which relates symbols in Wiener amalgam spaces to operators acting on classical modulation spaces.
Book chapters
2.
Bayer, Dominik; Cordero, Elena; Gröchenig, Karlheinz; Trapasso, S. Ivan
Linear perturbations of the Wigner transform and the Weyl quantization Book chapter
In: Advances in Microlocal and Time-Frequency Analysis, pp. 79–120, Birkhäuser, Cham, 2020, ISBN: 978-3-030-36137-2.
Abstract | Links | BibTeX | Tags: harmonic analysis, pseudodifferential operators, time-frequency analysis, time-frequency distributions, Wigner distributions
@incollection{BCGT_AMTFA_20,
title = {Linear perturbations of the Wigner transform and the Weyl quantization},
author = {Dominik Bayer and Elena Cordero and Karlheinz Gröchenig and S. Ivan Trapasso},
url = {https://arxiv.org/abs/1906.02503, arXiv},
doi = {10.1007/978-3-030-36138-9_5},
isbn = {978-3-030-36137-2},
year = {2020},
date = {2020-03-01},
urldate = {2020-03-01},
booktitle = {Advances in Microlocal and Time-Frequency Analysis},
pages = {79–120},
publisher = {Birkhäuser, Cham},
abstract = {We study a class of quadratic time-frequency representations that, roughly speaking, are obtained by linear perturbations of the Wigner transform. They satisfy Moyal's formula by default and share many other properties with the Wigner transform, but in general they do not belong to Cohen's class. We provide a characterization of the intersection of the two classes. To any such time-frequency representation, we associate a pseudodifferential calculus. We investigate the related quantization procedure, study the properties of the pseudodifferential operators, and compare the formalism with that of the Weyl calculus.},
keywords = {harmonic analysis, pseudodifferential operators, time-frequency analysis, time-frequency distributions, Wigner distributions},
pubstate = {published},
tppubtype = {incollection}
}
We study a class of quadratic time-frequency representations that, roughly speaking, are obtained by linear perturbations of the Wigner transform. They satisfy Moyal’s formula by default and share many other properties with the Wigner transform, but in general they do not belong to Cohen’s class. We provide a characterization of the intersection of the two classes. To any such time-frequency representation, we associate a pseudodifferential calculus. We investigate the related quantization procedure, study the properties of the pseudodifferential operators, and compare the formalism with that of the Weyl calculus.
1.
Trapasso, S. Ivan
Almost diagonalization of pseudodifferential operators Book chapter
In: Landscapes of Time-Frequency Analysis, pp. 323–342, Springer, 2019, ISBN: 978-3-030-05209-6.
Abstract | Links | BibTeX | Tags: almost diagonalization, harmonic analysis, pseudodifferential operators, time-frequency analysis
@incollection{T_LAND_19,
title = {Almost diagonalization of pseudodifferential operators},
author = {S. Ivan Trapasso},
doi = {10.1007/978-3-030-05210-2_14},
isbn = {978-3-030-05209-6},
year = {2019},
date = {2019-02-01},
urldate = {2019-02-01},
booktitle = {Landscapes of Time-Frequency Analysis},
pages = {323–342},
publisher = {Springer},
abstract = {In this review paper we focus on the almost diagonalization of pseudodifferential operators. We especially emphasize the advantages offered by a time-frequency analysis approach in connection with this issue.},
keywords = {almost diagonalization, harmonic analysis, pseudodifferential operators, time-frequency analysis},
pubstate = {published},
tppubtype = {incollection}
}
In this review paper we focus on the almost diagonalization of pseudodifferential operators. We especially emphasize the advantages offered by a time-frequency analysis approach in connection with this issue.