Convergence and Regularization of Sampling Series

This chapter consists of an overview of recent results concerning the convergence and regularization of the quintessential sampling series, the cardinal sine series. Conditions, that go beyond those associated with the standard theory, are formulated that ensure reconstruction by this series. The conditions are one of two types: (i) on the coefficients or samples or (ii) on the functions or signals being reconstructed. A class of regularization methods, that in effect consist of mollifications of cardinal sine series, is discussed. One of the highlights is a result that shows how piecewise polynomial splines and certain variants can be regarded as such mollifications.