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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todo:paper_10635203_v42_n3_p378_Aldroubi2023-10-03T16:01:12Z Dynamical sampling Aldroubi, A. Cabrelli, C. Molter, U. Tang, S. 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. JOUR info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar 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 Aldroubi, A. Cabrelli, C. Molter, U. Tang, S. 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. |
format |
JOUR |
author |
Aldroubi, A. Cabrelli, C. Molter, U. Tang, S. |
author_facet |
Aldroubi, A. Cabrelli, C. Molter, U. Tang, S. |
author_sort |
Aldroubi, A. |
title |
Dynamical sampling |
title_short |
Dynamical sampling |
title_full |
Dynamical sampling |
title_fullStr |
Dynamical sampling |
title_full_unstemmed |
Dynamical sampling |
title_sort |
dynamical sampling |
url |
http://hdl.handle.net/20.500.12110/paper_10635203_v42_n3_p378_Aldroubi |
work_keys_str_mv |
AT aldroubia dynamicalsampling AT cabrellic dynamicalsampling AT molteru dynamicalsampling AT tangs dynamicalsampling |
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1782024076227772416 |