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Sampling the flow of a bandlimited function

hal.structure.identifierDepartment of Mathematics, Vanderbilt University
dc.contributor.authorALDROUBI, Akram
hal.structure.identifierFakultät für Mathematik [Wien]
dc.contributor.authorGRÖCHENIG, Karlheinz
hal.structure.identifierDepartment of Mathematics [UCLA]
dc.contributor.authorHUANG, Longxiu
hal.structure.identifierInstitut de Mathématiques de Bordeaux [IMB]
dc.contributor.authorJAMING, Philippe
hal.structure.identifierNorthern Illinois University
dc.contributor.authorKRISHTAL, Ilya
hal.structure.identifierAcoustics Research Institute [ARI]
hal.structure.identifierFakultät für Mathematik [Wien]
dc.contributor.authorROMERO, José-Luis
dc.date.accessioned2024-04-04T02:54:58Z
dc.date.available2024-04-04T02:54:58Z
dc.date.issued2021
dc.identifier.issn1050-6926
dc.identifier.urihttps://oskar-bordeaux.fr/handle/20.500.12278/192321
dc.description.abstractEnWe analyze the problem of reconstruction of a bandlimited function $f$ from the space-time samples of its states $f_t=\phi_t\ast f$ resulting from the convolution with a kernel $\phi_t$. It is well-known that, in natural phenomena, uniform space-time samples of $f$ are not sufficient to reconstruct $f$ in a stable way. To enable stable reconstruction, a space-time sampling with periodic nonuniformly spaced samples must be used as was shown by Lu and Vetterli. We show that the stability of reconstruction, as measured by a condition number, controls the maximal gap between the spacial samples. We provide a quantitative statement of this result. In addition, instead of irregular space-time samples, % with irregular spatial sampling, we show that uniform dynamical samples at sub-Nyquist spatial rate allow one to stably reconstruct the function $\widehat f$ away from certain, explicitly described blind spots. We also consider several classes of finite dimensional subsets of bandlimited functions in which the stable reconstruction is possible, even inside the blind spots. We obtain quantitative estimates for it using Remez-Tur\'an type inequalities. En route, we obtain Remez-Tur\'an inequality for prolate spheroidal wave functions. To illustrate our results, we present some numerics and explicit estimates for the heat flow problem.
dc.language.isoen
dc.publisherSpringer
dc.title.enSampling the flow of a bandlimited function
dc.typeArticle de revue
dc.identifier.doi10.1007/s12220-021-00617-0
dc.subject.halMathématiques [math]/Analyse classique [math.CA]
dc.subject.halMathématiques [math]/Analyse fonctionnelle [math.FA]
dc.subject.halMathématiques [math]/Variables complexes [math.CV]
dc.subject.halSciences de l'ingénieur [physics]/Traitement du signal et de l'image
bordeaux.journalThe Journal of Geometric Analysis
bordeaux.page9241-9275
bordeaux.volume31
bordeaux.hal.laboratoriesInstitut de Mathématiques de Bordeaux (IMB) - UMR 5251*
bordeaux.institutionUniversité de Bordeaux
bordeaux.institutionBordeaux INP
bordeaux.institutionCNRS
bordeaux.peerReviewedoui
hal.identifierhal-02557625
hal.version1
hal.popularnon
hal.audienceInternationale
hal.origin.linkhttps://hal.archives-ouvertes.fr//hal-02557625v1
bordeaux.COinSctx_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.jtitle=The%20Journal%20of%20Geometric%20Analysis&rft.date=2021&rft.volume=31&rft.spage=9241-9275&rft.epage=9241-9275&rft.eissn=1050-6926&rft.issn=1050-6926&rft.au=ALDROUBI,%20Akram&GR%C3%96CHENIG,%20Karlheinz&HUANG,%20Longxiu&JAMING,%20Philippe&KRISHTAL,%20Ilya&rft.genre=article


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