This paper studies the problem of reconstructing continuous-time signals from discrete-time uniformly sampled data. This signal reconstruction problem has been studied by the authors in various contexts, and led to a new signal processing paradigm. The key idea there is to employ a physically realizable signal generator model, and design an (sub)optimal filter via H^ infty(mathbb C- +) optimal sampled-data control theory. The present paper aims at extending this framework to a more general setting where observed data are acquired through an acquisition device (prefilter) that has compact support. In this way, the framework can capture the properties of processing signals with a localized acquisition filter. We give a general setup as well as approximate solution methods along with their convergence results. A simulation is presented to illustrate some properties of the result.