System Representations for the Paley–Wiener Space $\mathcal {PW}_{\pi }^2$

Abstract

In this paper we study the approximation of stable linear time-invariant systems for the Paley–Wiener space $\mathcal {PW}_{\pi }^2$, i.e., the set of bandlimited functions with finite $L^2$-norm, by convolution sums. It is possible to use either, the convolution sum where the time variable is in the argument of the bandlimited impulse response, or the convolution sum where the time variable is in the argument of the function, as an approximation process. In addition to the pointwise and uniform convergence behavior, the convergence behavior in the norm of the considered function space, i.e. the $L^2$-norm in our case, is important. While it is well-known that both convolution sums converge uniformly on the whole real axis, the $L^2$-norm of the second convolution sum can be divergent for certain functions and systems. We show that the there exist an infinite dimensional closed subspace of functions and an infinite dimensional closed subspace of systems, such that for any pair of function and system from these two sets, we have norm divergence.