A general approach for convergence analysis of adaptive sampling-based signal processing

It is well-known that there exist bandlimited signals for which certain sampling series are divergent. One possible way of circumventing the divergence is to adapt the sampling series to the signals. In this paper we study adaptivity in the number of summands that are used in each approximation step, and whether this kind of adaptive signal processing can improve the convergence behavior of the sampling series. We approach the problem by considering approximation processes in general Banach spaces and show that adaptivity reduces the set of signals with divergence from a residual set to a meager or empty set. Due to the non-linearity of the adaptive approximation process, this study cannot be done by using the Banach-Steinhaus theory. We present examples from sampling based signal processing, where recently strong divergence, which is connected to the effectiveness of adaptive signal processing, has been observed.