Approximation of Classes of Convolutions by Linear Operators of Special Form

V. P. Zastavnyi ; V. V. Savchuk

Matematičeskie zametki, Tome 90 (2011) no. 3, pp. 351-361

Résumé

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 $p=\infty$.

Détail

Mots-clés : convolution, Euler polynomial
Keywords: linear operator, periodic measurable function, moment sequence, Borel measure, Fourier series, Bernoulli numbers.