Estimates of approximations by interpolation trigonometric polynomials on the classes of convolutions of high smoothness

Abstract

We establish interpolation analogues of Lebesgue type inequalities on the sets of $C^{\psi}_{\beta}L_{1}$ $2\pi$-periodic functions $f$, which are representable as convolutions of generating kernel $\Psi_{\beta}(t) = \sum\limits_{k=1}^{\infty}\psi(k)\cos \big(kt-\frac{\beta\pi}{2}\big)$, $\psi(k)\geq 0$, $\sum\limits_{k=1}^{\infty}\psi(k)<\infty$, $\beta\in\mathbb{R}$, with functions $\varphi$ from $L_{1}$ . In obtained inequalities for each $x\in\mathbb{R}$ the modules of deviations $|f(x)- \tilde{S}_{n-1}(f;x)|$ of interpolation Lagrange polynomials $ \tilde{S}_{n-1}(f;\cdot)$ are estimated via best approximations $E_{n}(\varphi)_{L_{1}}$ of functions $\varphi$ by trigonometric polynomials in $L_{1}$-metrics. When the sequences $\psi(k)$ decrease to zero faster than any power function, the obtained inequalities in many important cases are asymptotically exact. In such cases we also establish the asymptotic equalities for exact upper boundaries of pointwise approximations by interpolation trigonometric polynomials on the classes of convolutions of generating kernel $\Psi_{\beta}$ with functions $\varphi$, which belong to the unit ball from the space $L_{1}$.