On the existence of optimizers for time–frequency concentration problems

Fabio Nicola; José Luis Romero; S. Ivan Trapasso

doi:10.1007/s00526-022-02358-6

On the existence of optimizers for time–frequency concentration problems

Fabio Nicola, José Luis Romero, S. Ivan Trapasso

Department of Mathematics

Publications: Contribution to journal › Article › Peer Reviewed

Abstract

We consider the problem of the maximum concentration in a fixed measurable subset Ω ⊂ R²^d of the time-frequency space for functions f∈ L²(R^d). The notion of concentration can be made mathematically precise by considering the L^p-norm on Ω 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 Ω ⊂ R²^d of finite measure and every 1 ≤ p< ∞, there exists an optimizer for sup{‖A(f)‖Lp(Ω):f∈L2(Rd),‖f‖L2=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= ∞ 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(R^d) , 0 < q< 2 , equipped with continuous or discrete-type (quasi-)norms.

Original language	English
Article number	21
Journal	Calculus of Variations and Partial Differential Equations
Volume	62
Issue number	1
DOIs	https://doi.org/10.1007/s00526-022-02358-6
Publication status	Published - Jan 2023

Funding

The authors are very grateful to Karlheinz Gröchenig for bringing to their attention the problem solved here, in connection to an unpublished manuscript of his and Markus Neuhauser. The present research has been partially supported by the MIUR grant Dipartimenti di Eccellenza 2018–2022, CUP: E11G18000350001, DISMA, Politecnico di Torino. J. L. R. gratefully acknowledges support from the Austrian Science Fund (FWF): Y 1199. S. I. T. is member of the Machine Learning Genoa (MaLGa) Center, Università di Genova. F. N. and S. I. T. are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The authors are very grateful to Karlheinz Gröchenig for bringing to their attention the problem solved here, in connection to an unpublished manuscript of his and Markus Neuhauser. The present research has been partially supported by the MIUR grant Dipartimenti di Eccellenza 2018–2022, CUP: E11G18000350001, DISMA, Politecnico di Torino. J. L. R. gratefully acknowledges support from the Austrian Science Fund (FWF): Y 1199. S. I. T. is member of the Machine Learning Genoa (MaLGa) Center, Università di Genova. F. N. and S. I. T. are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

Austrian Fields of Science 2012

101002 Analysis

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10.1007/s00526-022-02358-6

Cite this

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title = "On the existence of optimizers for time–frequency concentration problems",

abstract = "We consider the problem of the maximum concentration in a fixed measurable subset Ω ⊂ R2d of the time-frequency space for functions f∈ L2(Rd). The notion of concentration can be made mathematically precise by considering the Lp-norm on Ω 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 Ω ⊂ R2d of finite measure and every 1 ≤ p< ∞, there exists an optimizer for sup\{‖A(f)‖Lp(Ω):f∈L2(Rd),‖f‖L2=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= ∞ and related optimization problems for the time correlation function, the cross-ambiguity function with a fixed window, and for functions in the modulation spaces Mq(Rd) , 0 < q< 2 , equipped with continuous or discrete-type (quasi-)norms.",

author = "Fabio Nicola and Romero, \{Jos{\'e} Luis\} and Trapasso, \{S. Ivan\}",

note = "Funding Information: The authors are very grateful to Karlheinz Gr{\"o}chenig for bringing to their attention the problem solved here, in connection to an unpublished manuscript of his and Markus Neuhauser. The present research has been partially supported by the MIUR grant Dipartimenti di Eccellenza 2018–2022, CUP: E11G18000350001, DISMA, Politecnico di Torino. J. L. R. gratefully acknowledges support from the Austrian Science Fund (FWF): Y 1199. S. I. T. is member of the Machine Learning Genoa (MaLGa) Center, Universit{\`a} di Genova. F. N. and S. I. T. are members of the Gruppo Nazionale per l{\textquoteright}Analisi Matematica, la Probabilit{\`a} e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). Funding Information: The authors are very grateful to Karlheinz Gr{\"o}chenig for bringing to their attention the problem solved here, in connection to an unpublished manuscript of his and Markus Neuhauser. The present research has been partially supported by the MIUR grant Dipartimenti di Eccellenza 2018–2022, CUP: E11G18000350001, DISMA, Politecnico di Torino. J. L. R. gratefully acknowledges support from the Austrian Science Fund (FWF): Y 1199. S. I. T. is member of the Machine Learning Genoa (MaLGa) Center, Universit{\`a} di Genova. F. N. and S. I. T. are members of the Gruppo Nazionale per l{\textquoteright}Analisi Matematica, la Probabilit{\`a} e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). Publisher Copyright: {\textcopyright} 2022, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.",

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AU - Romero, José Luis

AU - Trapasso, S. Ivan

N1 - Funding Information: The authors are very grateful to Karlheinz Gröchenig for bringing to their attention the problem solved here, in connection to an unpublished manuscript of his and Markus Neuhauser. The present research has been partially supported by the MIUR grant Dipartimenti di Eccellenza 2018–2022, CUP: E11G18000350001, DISMA, Politecnico di Torino. J. L. R. gratefully acknowledges support from the Austrian Science Fund (FWF): Y 1199. S. I. T. is member of the Machine Learning Genoa (MaLGa) Center, Università di Genova. F. N. and S. I. T. are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). Funding Information: The authors are very grateful to Karlheinz Gröchenig for bringing to their attention the problem solved here, in connection to an unpublished manuscript of his and Markus Neuhauser. The present research has been partially supported by the MIUR grant Dipartimenti di Eccellenza 2018–2022, CUP: E11G18000350001, DISMA, Politecnico di Torino. J. L. R. gratefully acknowledges support from the Austrian Science Fund (FWF): Y 1199. S. I. T. is member of the Machine Learning Genoa (MaLGa) Center, Università di Genova. F. N. and S. I. T. are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). Publisher Copyright: © 2022, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

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Y1 - 2023/1

N2 - We consider the problem of the maximum concentration in a fixed measurable subset Ω ⊂ R2d of the time-frequency space for functions f∈ L2(Rd). The notion of concentration can be made mathematically precise by considering the Lp-norm on Ω 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 Ω ⊂ R2d of finite measure and every 1 ≤ p< ∞, there exists an optimizer for sup{‖A(f)‖Lp(Ω):f∈L2(Rd),‖f‖L2=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= ∞ and related optimization problems for the time correlation function, the cross-ambiguity function with a fixed window, and for functions in the modulation spaces Mq(Rd) , 0 < q< 2 , equipped with continuous or discrete-type (quasi-)norms.

AB - We consider the problem of the maximum concentration in a fixed measurable subset Ω ⊂ R2d of the time-frequency space for functions f∈ L2(Rd). The notion of concentration can be made mathematically precise by considering the Lp-norm on Ω 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 Ω ⊂ R2d of finite measure and every 1 ≤ p< ∞, there exists an optimizer for sup{‖A(f)‖Lp(Ω):f∈L2(Rd),‖f‖L2=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= ∞ and related optimization problems for the time correlation function, the cross-ambiguity function with a fixed window, and for functions in the modulation spaces Mq(Rd) , 0 < q< 2 , equipped with continuous or discrete-type (quasi-)norms.

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DO - 10.1007/s00526-022-02358-6

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JO - Calculus of Variations and Partial Differential Equations

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ER -

On the existence of optimizers for time–frequency concentration problems

Abstract

Funding

Austrian Fields of Science 2012

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