Title :
Sampling and Interpolation Problems for Vector Valued Signals in the Paley–Wiener Spaces
Author :
Avdonin, Sergei A. ; Ivanov, Sergei A.
Author_Institution :
Dept. of Math. & Stat., Alaska Univ., Fairbanks, AK
Abstract :
A new approach to sampling/interpolation problem for vector valued signals is proposed. For a function (signal) f from the Paley-Wiener space PW pi(BBC N) of entire vector valued functions, the sampling data are defined as the inner products langf(lambdan), alphan rang of f(lambdan) and vectors alphan. It is proved that { alphan, lambdan} is a sampling and interpolating sequence for PW pi(BBC N) if and only if the family { alphanei lambdan t} forms a Riesz basis in L 2(-pi,pi; BBC N). This relation allows to characterize all sampling and interpolating sequences in terms of a generating matrix function. To produce a wide class of matrix functions generating sampling and interpolating sequences we consider the matrix Sturm-Liouville problems and apply the boundary control theory for hyperbolic dynamical systems. This approach allows to construct both asymptotically uniform and nonuniform sampling and interpolating sequences.
Keywords :
Sturm-Liouville equation; interpolation; sequences; signal sampling; Paley-Wiener spaces; Riesz basis; hyperbolic dynamical systems; interpolation; matrix Sturm-Liouville problems; sampling; vector valued signals; Controllability; Riesz bases; Sampling; Sturm–Liouville problem; Sturm-Liouville problem; controllability; families of exponentials; interpolation; sampling;
Journal_Title :
Signal Processing, IEEE Transactions on
DOI :
10.1109/TSP.2008.928702