Sampling and Interpolation Problems for Vector Valued Signals in the Paley–Wiener Spaces

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.