Multilevel and Progressive Iterative Methods for Approximation and Numerical Integration

Recently two kinds of approximation techniques have been introduced and studied in the literature. As far as we know, both seminal papers [1, 2], one dealing with the socalled multilevel approximation (MA) and the other one with the so-called progressive iterative approximation (PIA), were published in 2004. They are two different ways of triggering an iterative procedure, acting on some kind of remainder (or error) in the base method. Induced by these ideas, we are interested in investigating and comparing both techniques MA and PIA, when applied to spline QI operators on bounded domains, in particular to the well known variation-diminishing Schoenberg-Marsden operator, suitably modified in order to satisfy the main interest that lies in providing best approximation order, while being easy to compute. This study is also aimed at defining new quadrature formulas, based on such iterative techniques.