DURRMEYER SAMPLING TYPE OPERATORS IN FUNCTIONAL SPACES: A UNIFYING STUDY INTO REGULARIZATION, CONVERGENCE AND ORDER OF APPROXIMATION

This thesis aims to provide a unifying study on the approximation properties, including regularization results, for a semi-discrete version of sampling operators, represented by the so-called Durrmeyer sampling type operators. We study their regularization properties by using a distributional approach, we provide convergence results in several functional spaces, and we estimate the order of approximation via a quantitative and qualitative analysis. To this aim, we use a unifying method that leads to approximation results covering a wide range of functions, including those not necessarily continuous, so as to obtain advantages from both an applications and theoretical point of view. The main framework is indeed represented by the general setting of Orlicz spaces, introduced as a natural extension of the well-known Lebesgue spaces, that include several functional spaces as particular cases. Furthermore, in order to open the way for possible future applications of the theory to the Image Processing, we also extend the main approximation properties presented here in the multidimensional setting.