Dynamical sampling

• Autores: Aldroubi, A.; Cabrelli, Carlos; Molter, Ursula Maria; Tang, S.
• Año de publicación: 2017
• Idioma: inglés
• Tipo de recurso: artículo
• Materia: Carleson'S Theorem; Feichtinger Conjecture; Frames; Müntz–Szász Theorem; Reconstruction; Sampling Theory; Sub-Sampling
• Institución: Consejo Nacional de Investigaciones Científicas y Técnicas

Let Y={f(i),Af(i),…,Ali f(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.
Fil: Aldroubi, A.. Vanderbilt University; Estados Unidos
Fil: Cabrelli, Carlos. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina
Fil: Molter, Ursula Maria. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina
Fil: Tang, S.. Vanderbilt University; Estados Unidos