Dynamical sampling

Let Y={f(i),Af(i),…,Alif(i):i∈Ω}, where A is a bounded operator on ℓ2(I). The problem under consideration is to find necessary and sufficient conditions on A,Ω,{li:i∈Ω} in order to recover any f∈ℓ2(I) from the measurements Y. This is the so-called dynamical sampling problem in which we seek to recov...

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Autores principales:	Cabrelli, Carlos Alberto, Molter, Ursula Maria
Publicado:	2017
Materias:	Carleson's theorem Feichtinger conjecture Frames Müntz–Szász Theorem Reconstruction Sampling theory Sub-sampling Harmonic analysis Image reconstruction Problem solving
Acceso en línea:	https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_10635203_v42_n3_p378_Aldroubi http://hdl.handle.net/20.500.12110/paper_10635203_v42_n3_p378_Aldroubi
Aporte de:	Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires

id	paper:paper_10635203_v42_n3_p378_Aldroubi
record_format	dspace
spelling	paper:paper_10635203_v42_n3_p378_Aldroubi2023-06-08T16:03:32Z Dynamical sampling Cabrelli, Carlos Alberto Molter, Ursula Maria Carleson's theorem Feichtinger conjecture Frames Müntz–Szász Theorem Reconstruction Sampling theory Sub-sampling Harmonic analysis Image reconstruction Carleson's theorem Feichtinger conjecture Frames Sampling theory Sub-sampling Problem solving Let Y={f(i),Af(i),…,Alif(i):i∈Ω}, where A is a bounded operator on ℓ2(I). The problem under consideration is to find necessary and sufficient conditions on A,Ω,{li:i∈Ω} in order to recover any f∈ℓ2(I) from the measurements Y. This is the so-called dynamical sampling problem in which we seek to recover a function f by combining coarse samples of f and its futures states Alf. We completely solve this problem in finite dimensional spaces, and for a large class of self adjoint operators in infinite dimensional spaces. In the latter case, although Y can be complete, using the Müntz–Szász Theorem we show it can never be a basis. We can also show that, when Ω is finite, Y is not a frame except for some very special cases. The existence of these special cases is derived from Carleson's Theorem for interpolating sequences in the Hardy space H2(D). Finally, using the recently proved Kadison–Singer/Feichtinger theorem we show that the set obtained by normalizing the vectors of Y can never be a frame when Ω is finite. © 2015 Elsevier Inc. Fil:Cabrelli, C. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. Fil:Molter, U. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. 2017 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_10635203_v42_n3_p378_Aldroubi http://hdl.handle.net/20.500.12110/paper_10635203_v42_n3_p378_Aldroubi
institution	Universidad de Buenos Aires
institution_str	I-28
repository_str	R-134
collection	Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
topic	Carleson's theorem Feichtinger conjecture Frames Müntz–Szász Theorem Reconstruction Sampling theory Sub-sampling Harmonic analysis Image reconstruction Carleson's theorem Feichtinger conjecture Frames Sampling theory Sub-sampling Problem solving
spellingShingle	Carleson's theorem Feichtinger conjecture Frames Müntz–Szász Theorem Reconstruction Sampling theory Sub-sampling Harmonic analysis Image reconstruction Carleson's theorem Feichtinger conjecture Frames Sampling theory Sub-sampling Problem solving Cabrelli, Carlos Alberto Molter, Ursula Maria Dynamical sampling
topic_facet	Carleson's theorem Feichtinger conjecture Frames Müntz–Szász Theorem Reconstruction Sampling theory Sub-sampling Harmonic analysis Image reconstruction Carleson's theorem Feichtinger conjecture Frames Sampling theory Sub-sampling Problem solving
description	Let Y={f(i),Af(i),…,Alif(i):i∈Ω}, where A is a bounded operator on ℓ2(I). The problem under consideration is to find necessary and sufficient conditions on A,Ω,{li:i∈Ω} in order to recover any f∈ℓ2(I) from the measurements Y. This is the so-called dynamical sampling problem in which we seek to recover a function f by combining coarse samples of f and its futures states Alf. We completely solve this problem in finite dimensional spaces, and for a large class of self adjoint operators in infinite dimensional spaces. In the latter case, although Y can be complete, using the Müntz–Szász Theorem we show it can never be a basis. We can also show that, when Ω is finite, Y is not a frame except for some very special cases. The existence of these special cases is derived from Carleson's Theorem for interpolating sequences in the Hardy space H2(D). Finally, using the recently proved Kadison–Singer/Feichtinger theorem we show that the set obtained by normalizing the vectors of Y can never be a frame when Ω is finite. © 2015 Elsevier Inc.
author	Cabrelli, Carlos Alberto Molter, Ursula Maria
author_facet	Cabrelli, Carlos Alberto Molter, Ursula Maria
author_sort	Cabrelli, Carlos Alberto
title	Dynamical sampling
title_short	Dynamical sampling
title_full	Dynamical sampling
title_fullStr	Dynamical sampling
title_full_unstemmed	Dynamical sampling
title_sort	dynamical sampling
publishDate	2017
url	https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_10635203_v42_n3_p378_Aldroubi http://hdl.handle.net/20.500.12110/paper_10635203_v42_n3_p378_Aldroubi
work_keys_str_mv	AT cabrellicarlosalberto dynamicalsampling AT molterursulamaria dynamicalsampling
_version_	1768546502926401536

Dynamical sampling

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