Approximation of Classes of Convolutions by Linear Operators of Special Form

A parametric family of operators $G_\rho$ is constructed for the class of convolutions $\mathbf{W}_{p,m}(K)$ whose kernel $K$ was generated by the moment sequence. We obtain a formula for evaluating
$$ E(\mathbf{W}_{p,m}(K);G_\rho)_p:=\sup_{f\in\mathbf{W}_{p,m}(K)}\|f-G_\rho(f)\|_p. $$
For the case in which $\mathbf{W}_{p,m}(K)=\mathbf{W}^{r,\beta}_{p,m}$, we obtain an expansion in powers of the parameter $\varepsilon=-\ln\rho$ for $E(\mathbf{W}^{r,\beta}_{p,m};G_{\rho,r})_p$, where $\beta\in\mathbb{Z}$, $r>0$, and $m\in\mathbb{N}$, while $p=1$ or $p=\infty$.