Donoho-Logan Large Sieve Principles for Modulation and Polyanalytic Fock\n Spaces

2018/08/07 by Luís Daniel Abreu, Luis Daniel Abreu, Michael Speckbacher +2 · 5 citations
Computer Science · Engineering · Mathematics · #11N36 #42C15 #42C40 #46E15 #46E20 #Algorithm #Combinatorics #FOS: Mathematics #Fock space #Fourier analysis #Fourier transform #Function (biology) #Functional Analysis (math.FA) #Hermite polynomials #Mathematical Analysis and Transform Methods #Mathematical analysis #Mathematics #Physics #Pure mathematics #Quantum mechanics #Short-time Fourier transform #Sparse and Compressive Sensing Techniques #Topology (electrical circuits) #math.FA #msc:11N36 #msc:42C15 #msc:42C40 #msc:46E15 #msc:46E20

paper · pdf · doi:10.48550/arxiv.1808.02258

published in arXiv (Cornell University) (Cornell University)

arxiv created 2018/08/07 · openalex publication_date 2018/08/07 · arxiv updated 2018/08/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We obtain estimates for the L^p-norm of the short-time Fourier transform\n(STFT) for functions in modulation spaces, providing information about the\nconcentration on a given subset of \ℝ², leading to deterministic\nguarantees for perfect reconstruction using convex optimization methods. More\nprecisely, we will obtain large sieve inequalities of the Donoho-Logan type,\nbut instead of localizing the signals in regions T\× W of the\ntime-frequency plane using the Fourier transform to intertwine time and\nfrequency, we will localize the representation of the signals in terms of the\nshort-time Fourier transform in sets \Δ with arbitrary geometry. At the\ntechnical level, since there is no proper analogue of Beurling's extremal\nfunction in the STFT setting, we introduce a new method, which rests on a\ncombination of an argument similar to Schur's test with an extension of Seip's\nlocal reproducing formula to general Hermite windows. When the windows are\nHermite functions, we obtain local reproducing formulas for polyanalytic Fock\nspaces which lead to explicit large sieve constant estimates and, as a\nbyproduct, to a reconstruction formula for f\∈ L²(\ℝ) from its\nSTFT values on arbitrary discs. A discussion on optimality follows, along the\nlines of Donoho-Stark paper on uncertainty principles and signal recovery. We\nalso consider the case of discrete Gabor systems, vector-valued STFT transforms\nand rephrase the results in terms of the polyanalytic Bargmann-Fock transforms.\n

Citations

• Compressed sensing
• Asymptotic expansion of polyanalytic Bergman kernels

Cited by

• On quantum large sieve inequalities and operator recovery from incomplete information
• The isoperimetric inequality for the Ky Fan norm
• Sharp stability of convex functionals on weighted Bergman spaces with applications
• Planar sampling sets for the short-time Fourier transform
• Daubechies' Time-Frequency Localization Operator on Cantor Type Sets

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• On quantum large sieve inequalities and operator recovery from incomplete information
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