Rate of approximation for nonlinear integral operators with application to signal processing

Abstract

We study the problem of the order of approximation in modular spaces for a family of nonlinear integral operators of the form $$ (T_wf)(s) = \displaystyle\int_H K_w (s- h_w(t), f(h_w(t))) d \mu_H (t),~~w>0,~ s \in G, $$ where $G$~and $H$~ are locally compact topological groups, $f:G{\mbox{$\rightarrow$}}{\mathbb R}$ is measurable, $\{ h_w \}_{w>0}$~is a family of homeomorphisms $h_w:H \rightarrow h_w(H) \subset G$ and $\{K_w\}_{w>0}$ is a family of kernel functions. The general setting of modular spaces allows us to obtain, in particular, the rate of approximation for the above operators in $L^p$ spaces and in Orlicz-type spaces. Furthermore, the above general class contains, as particular cases, some classical families of integral operators well known in approximation theory, such as the classical convolution integral operators, the Mellin convolution integral operators and the sampling-type operators in their nonlinear form. Our approach, in the framework of modular spaces, is mainly based on the introduction of a suitable Lipschitz class and of a condition on a family of measures which is linked with the modulars involved and which is always fulfilled in classical and Musielak-Orlicz spaces.