Dynamical sampling

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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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Autor principal: Aldroubi, A.
Otros Autores: Cabrelli, C., Molter, U., Tang, S.
Formato: Capítulo de libro
Lenguaje:Inglés
Publicado: Academic Press Inc. 2017
Acceso en línea:Registro en Scopus
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100 1 |a Aldroubi, A. 
245 1 0 |a Dynamical sampling 
260 |b Academic Press Inc.  |c 2017 
270 1 0 |m Aldroubi, A.; Department of Mathematics, Vanderbilt UniversityUnited States; email: aldroubi@math.vanderbilt.edu 
506 |2 openaire  |e Política editorial 
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520 3 |a 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.  |l eng 
593 |a Department of Mathematics, Vanderbilt University, Nashville, TN 37240-0001, United States 
593 |a Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, Pabellón I, Buenos Aires, 1428, Argentina 
593 |a CONICET, Consejo Nacional de Investigaciones Científicas y Técnicas, Argentina 
690 1 0 |a CARLESON'S THEOREM 
690 1 0 |a FEICHTINGER CONJECTURE 
690 1 0 |a FRAMES 
690 1 0 |a MUNTZ–SZASZ THEOREM 
690 1 0 |a RECONSTRUCTION 
690 1 0 |a SAMPLING THEORY 
690 1 0 |a SUB-SAMPLING 
690 1 0 |a HARMONIC ANALYSIS 
690 1 0 |a IMAGE RECONSTRUCTION 
690 1 0 |a CARLESON'S THEOREM 
690 1 0 |a FEICHTINGER CONJECTURE 
690 1 0 |a FRAMES 
690 1 0 |a SAMPLING THEORY 
690 1 0 |a SUB-SAMPLING 
690 1 0 |a PROBLEM SOLVING 
700 1 |a Cabrelli, C. 
700 1 |a Molter, U. 
700 1 |a Tang, S. 
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