Convergence and rate of approximation for linear integral operators in \(BV^{\varphi }\)-spaces in multidimensional setting
The authors introduce the space \(BV^{\varphi}(\mathbb{R}^N)\) of functions of bounded \(\varphi\)-variation in the sense of Tonelli and the \(\varphi\)-modulus of smoothness \(\omega_{\varphi}(f,\delta)\) for the functions \(f\) defined on \(\mathbb{R}^N\). For the linear convolution operators \[ (T_wf)(s)=\int_{\mathbb{R}^N}K_w(t)f(s-t)\,dt, \quad w>0, \quad s\in \mathbb{R}^N, \] under classical singularity assumptions on the family of kernels \(\{K_w\}_{w>0}\) the convergence results and rate of covergence results in the space \(BV^{\varphi}(\mathbb{R}^N)\) or in some of its subspaces are established.
scientific article; zbMATH DE number 5362159
scientific article