General behavior of sampling-based signal and system representation

We analyze sampling representations for translation invariant, linear and bounded systems, operating on band-limited signals. First, we characterize suitable kernels for reconstruction processes with and without oversampling. Then, we investigate the convergence behavior of general approximation processes, operating only on the samples and not on the whole continuous-time signal, for translation invariant, linear and bounded systems and signals in the Paley-Wiener space PW_pi ¹. Recently, Habib analyzed similar questions for a larger space of functions, namely the Zakai class, but for a considerably smaller class of systems, not including the Hilbert transformation and the ideal low-pass filter. We show that for important systems there exists no approximation process that is uniformly convergent for all functions in PW_pi ¹. Surprisingly, oversampling and the design of special kernels does not improve the convergence behavior in this case. Furthermore, a simple criterion is given for checking whether a certain approximation process is convergent for a given system or not.