Uniform convergence and rate of approximation for a nonlinear version of the generalized sampling operator
Let \(G\) and \(H\) be locally compact (unimodular) topological groups, \(\mu_H\) -- the Haar measure on \(H\), and let \(h_w: H\to G\) be a homeomorphism between \(H\) and \(h_w(H)\), \(w\in\mathbb{R}\). There are investigated linear integral operators \(T_w\) of the form \[ (T_wf)(s)= \int_H K_w(s- h_w(t), f(h_w(t)) d\mu_H(t),\quad w\in \mathbb{R}, \] for \(f\in C(\mathbb{R})\). There are obtained results concerning approximation of \(f\) by \(T_wf\) in \(C(\mathbb{R})\), with specifying the order of approximation. Application is given in the case of nonlinear sampling operators. A large list of references is added.
scientific article; zbMATH DE number 1267808
scientific article