Study of dynamical sampling and frame representations

Abstract

Let Y =(i)X(i) Xmi(i) : HXB(H)andi,whereHisa separable Hilbert space. Our main objective is to establish the essential require ments and criteria involving X retrieve any function mi : i that are necessary and sucient to Hfrom the given sample set YThis is known as the dy namical sampling problem. Our objective is to reconstruct by combining rough samples of with its future states XDynamical sampling is widely applicable in various domains, including time-space sampling trade-o, super-resolution, on chip sensing, satellite remote sensing, and more. In nite-dimensional spaces, we discuss this problem for a diverse class of bounded linear operators, including di agonalizable, convolution, Fourier multiplier, and translation invariant operators. Next, we analyze when a frame nN n=1 is dynamical frame for 2(Zd), i.e. there exist a bounded linear operator X : 2(Zd) 2 (Zd) and 2 (Zd) such that nN n=1 = XnN n=1. We provide characterization results for dynamical frame and dynamical dual frame. Moreover, we demonstrate that each overcomplete frame possesses an innite number of dynamical dual frames.