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

TL;DR

This paper develops new inequalities for approximating smooth periodic functions using interpolation trigonometric polynomials, linking approximation errors to best approximations of related functions, with asymptotic exactness in certain cases.

Contribution

It introduces interpolation analogues of Lebesgue inequalities for classes of convolutions with high smoothness, providing asymptotically exact bounds for approximation errors.

Findings

Derived interpolation inequalities relating deviation modules to best approximations.

Established asymptotic equalities for pointwise approximation boundaries.

Proved asymptotic exactness when the kernel decay exceeds any power function.

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 $L_{1}$.