Optimization properties of generalized Chebyshev-Poisson integrals (Q6610551)

In this paper, an extremal problem of theory of approximating functions defined on a segment by linear methods of summing Fourier series is solved. Namely, it is raised the issue on the approximation of functions from the class \(H^1\) by generalized Chebyshev-Poisson integrals\N\[\N\begin{multlined}\NP_{s,q}(\rho;f;x)=\frac1\pi \int_{-1}^1\Big\{1+\sum_{k=1}^{+\infty}\rho^{k}[1+sk(1+\rho)(1-\rho)^{q}]\cos k(\arccos x-\arccos y)\\\N+\sum_{k=1}^{+\infty}\rho^{k}[1+sk(1+\rho)(1-\rho)^{q}]\cos k(\arccos x+\arccos y)\Big\}\frac{f(y)dy}{\sqrt{1-y^2}},\N\end{multlined}\tag{1}\N\]\Nwhere \(0\leq s\leq 1/2\) and \(q\geq 1\). \N\NIn particular, the authors established an exact equality for the quantity\N\[\N\mathcal{E}(H^1;P_{s,q}(\rho;x))=\sup_{f\in H^1} |f(x)-P_{s,q}(\rho;f;x)|.\tag{2}\N\]\NThe positive operator $(1)$ has a number of significant advantages in comparison with Abel-Poisson integrals and biharmonic Poisson integrals, which also belong to the class of positive operators. After all, the dependence of the Chebyshev-Poisson operator $(1)$ on two additional parameters \(s\) and \(q\) opens up wide opportunities for investigating the optimality of a solution to an extremal problem of the type $(2)$. \N\NThe results obtained in the present research summarize the results established in [\textit{J. I. Rusecki}, Sib. Mat. Zh. 9, 136--144 (1968; Zbl 0172.34102)] etc.