Publications
Preprints
2.
Svela, Erling; Trapasso, S. Ivan
A quantum harmonic analysis approach to nonlinear time-frequency concentration Preprint
2026.
Abstract | Links | BibTeX | Tags: concentration problems, quantum harmonic analysis, time-frequency analysis, time-frequency distributions
@workingpaper{ST_26,
title = {A quantum harmonic analysis approach to nonlinear time-frequency concentration},
author = {Erling Svela and S. Ivan Trapasso},
url = {https://arxiv.org/abs/2605.28786, arXiv},
year = {2026},
date = {2026-05-28},
abstract = {We study nonlinear concentration problems for time-frequency distributions in the Cohen class. Using recent techniques from quantum harmonic analysis (QHA) we provide both positive and negative results, such as sufficient conditions for the existence of optimizers in terms of the ``window operator'' and explicit examples where the supremum is never attained. We also study the structural properties of window operators, in particular operators that yield weakly continuous concentration functionals and operators for which the nonlinear concentration problem admits an optimizer, also beyond the Heisenberg representation. We then consider generalizations to the study of concentration problems for phase space representations of operators. We consider generalized Husimi distributions via quantum convolution, and their optimization problem when optimizing over Hilbert--Schmidt and density operators. Lastly, we consider representations of operators on double phase space, in the spirit of quantum time-frequency analysis, and give a full solution in terms of the Weyl symbols.},
keywords = {concentration problems, quantum harmonic analysis, time-frequency analysis, time-frequency distributions},
pubstate = {published},
tppubtype = {workingpaper}
}
We study nonlinear concentration problems for time-frequency distributions in the Cohen class. Using recent techniques from quantum harmonic analysis (QHA) we provide both positive and negative results, such as sufficient conditions for the existence of optimizers in terms of the “window operator” and explicit examples where the supremum is never attained. We also study the structural properties of window operators, in particular operators that yield weakly continuous concentration functionals and operators for which the nonlinear concentration problem admits an optimizer, also beyond the Heisenberg representation. We then consider generalizations to the study of concentration problems for phase space representations of operators. We consider generalized Husimi distributions via quantum convolution, and their optimization problem when optimizing over Hilbert–Schmidt and density operators. Lastly, we consider representations of operators on double phase space, in the spirit of quantum time-frequency analysis, and give a full solution in terms of the Weyl symbols.
1.
Stra, Federico; Svela, Erling; Trapasso, S. Ivan
2026.
Abstract | Links | BibTeX | Tags: Born-Jordan quantization, concentration problems, quantum harmonic analysis, time-frequency analysis, time-frequency distributions
@workingpaper{SST_26,
title = {On the existence of optimizers for nonlinear time-frequency concentration problems: the Born-Jordan distribution},
author = {Federico Stra and Erling Svela and S. Ivan Trapasso},
url = {https://arxiv.org/abs/2605.10535, arXiv},
year = {2026},
date = {2026-05-12},
urldate = {2026-05-12},
abstract = {We study the (L^p) concentration problem for the Born--Jordan distribution in dimension (d>1), thus extending the one-dimensional analysis in [Stra-Svela-Trapasso, J. Math. Pures Appl. (2026)]. We show that the existence of concentration optimizers depends on the exponent (p) with a critical threshold at (p_*(d)= frac{2d}{d-2}) for (dgeq2) (with the understanding that (p_*(2)=infty)). In particular, for subcritical exponents (1leq p<p_*(d)) we prove that the supremum is finite and is attained, whereas for supercritical exponents (p>p_*(d)) we show that the functional is unbounded. We also provide the complete solution in the (significantly more) challenging critical regime in dimension (d=2).},
keywords = {Born-Jordan quantization, concentration problems, quantum harmonic analysis, time-frequency analysis, time-frequency distributions},
pubstate = {published},
tppubtype = {workingpaper}
}
We study the \(L^p\) concentration problem for the Born–Jordan distribution in dimension \(d>1\), thus extending the one-dimensional analysis in [Stra-Svela-Trapasso, J. Math. Pures Appl. (2026)]. We show that the existence of concentration optimizers depends on the exponent \(p\) with a critical threshold at \(p_*(d)= \frac{2d}{d-2}\) for \(d\geq2\) (with the understanding that \(p_*(2)=\infty\)). In particular, for subcritical exponents \(1\leq p<p_*(d)\) we prove that the supremum is finite and is attained, whereas for supercritical exponents \(p>p_*(d)\) we show that the functional is unbounded. We also provide the complete solution in the (significantly more) challenging critical regime in dimension \(d=2\).
Journal Articles
3.
Stra, Federico; Svela, Erling; Trapasso, S. Ivan
On the existence of optimizers for nonlinear time-frequency concentration problems: the Wigner distribution Journal Article
In: J. Math. Pures Appl., vol. 201, pp. 103919, 2026.
Abstract | Links | BibTeX | Tags: concentration problems, quantum harmonic analysis, time-frequency analysis, time-frequency distributions, Wigner distributions
@article{SST_25,
title = {On the existence of optimizers for nonlinear time-frequency concentration problems: the Wigner distribution},
author = {Federico Stra and Erling Svela and S. Ivan Trapasso},
url = {https://arxiv.org/abs/2510.18683, arXiv
https://www.sciencedirect.com/science/article/pii/S0021782426000656/pdfft?md5=6dd623c3313b1a76deecf9a455f9f414&pid=1-s2.0-S0021782426000656-main.pdf, PDF},
doi = {10.1016/j.matpur.2026.103919},
year = {2026},
date = {2026-03-25},
urldate = {2026-03-25},
journal = {J. Math. Pures Appl.},
volume = {201},
pages = {103919},
abstract = {We prove that, for any measurable phase space subset (Omegasubsetmathbb{R}^{2d}) with (0<|Omega|<infty) and any (1le p < infty), the nonlinear concentration problem [sup_{f in L^2(mathbb{R}^d)setminus{0}}frac{|Wf|_{L^p(Omega)}}{|f|_{L^2}^2}] admits an optimizer, where (Wf) is the Wigner distribution of (f). The main obstruction is that (Wf) is covariant (not invariant) under time-frequency shifts, which impedes weak upper semicontinuity, so the effects of constructive interference must be taken into account. We close this compactness gap via concentration compactness for Heisenberg-type dislocations, together with a new asymptotic formula that quantifies the limiting contribution to concentration over (Omega) from asymptotically separated wave packets. When (p=infty) we also identify the sharp constant (2^d) and show that it is attained. We also discuss some related extensions: For (tau)-Wigner distributions with (tau in (0,1)) we isolate a chain phenomenon that obstructs the same strategy beyond the Wigner case ((tau=1/2)), while for the Born-Jordan distribution in (d=1) we obtain weak continuity, and thus existence of concentration optimizers for all (1le p<infty) (the (p=infty) supremum equals (pi) but is not attained).},
keywords = {concentration problems, quantum harmonic analysis, time-frequency analysis, time-frequency distributions, Wigner distributions},
pubstate = {published},
tppubtype = {article}
}
We prove that, for any measurable phase space subset \(\Omega\subset\mathbb{R}^{2d}\) with \(0<|\Omega|<\infty\) and any \(1\le p < \infty\), the nonlinear concentration problem \[\sup_{f \in L^2(\mathbb{R}^d)\setminus\{0\}}\frac{\|Wf\|_{L^p(\Omega)}}{\|f\|_{L^2}^2}\] admits an optimizer, where \(Wf\) is the Wigner distribution of \(f\). The main obstruction is that \(Wf\) is covariant (not invariant) under time-frequency shifts, which impedes weak upper semicontinuity, so the effects of constructive interference must be taken into account. We close this compactness gap via concentration compactness for Heisenberg-type dislocations, together with a new asymptotic formula that quantifies the limiting contribution to concentration over \(\Omega\) from asymptotically separated wave packets. When \(p=\infty\) we also identify the sharp constant \(2^d\) and show that it is attained. We also discuss some related extensions: For \(\tau\)-Wigner distributions with \(\tau \in (0,1)\) we isolate a chain phenomenon that obstructs the same strategy beyond the Wigner case (\(\tau=1/2\)), while for the Born-Jordan distribution in \(d=1\) we obtain weak continuity, and thus existence of concentration optimizers for all \(1\le p<\infty\) (the \(p=\infty\) supremum equals \(\pi\) but is not attained).
2.
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.
1.
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.
Book chapters
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.