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 crux there is to employ a physically realizable signal generator model, and design an (sub)optimal filter via HTC(C+) optimal sampled-data control theory. The present paper extends this framework to the situation where sampling is more general having a generalized sampling kernel. It is more consistent with a more general framework, for example, wavelet signal expansion, and can lead to a more general applications. We give a general setup along with a solution via fast-sample/fast-hold approximation. A simulation is presented to illustrate the result.