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Sampling the flow of a bandlimited function

Sampling the flow of a bandlimited function

Authors :
Longxiu Huang
Karlheinz Gröchenig
Philippe Jaming
Akram Aldroubi
Ilya A. Krishtal
José Luis Romero
Department of Mathematics, Vanderbilt University
Vanderbilt University [Nashville]
Fakultät für Mathematik [Wien]
Universität Wien
Department of Mathematics [UCLA]
University of California [Los Angeles] (UCLA)
University of California-University of California
Institut de Mathématiques de Bordeaux (IMB)
Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)
Northern Illinois University
Acoustics Research Institute (ARI)
Austrian Academy of Sciences (OeAW)
Publication Year :
2020
Publisher :
HAL CCSD, 2020.

Abstract

We 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, 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 a 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.<br />Comment: 29 pages

Details

Language :
English
Database :
OpenAIRE
Accession number :
edsair.doi.dedup.....4941fbb2f4f4c29a25b13e3d4c33516a