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 lang
f
(lambda
n
), alpha
n
rang of
f
(lambda
n
) and vectors alpha
n
. It is proved that { alpha
n
, lambda
n
} is a sampling and interpolating sequence for
PW

_pi
(\BBC
N
) if and only if the family { alpha
nei
lambda
n

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.