Approximation by interpolation trigonometric polynomials in metrics of the spaces $L_p$ on the classes of periodic entire functions

TL;DR

This paper derives asymptotic bounds for the approximation of periodic functions by interpolation trigonometric polynomials in $L_p$ spaces, focusing on functions representable as convolutions with specific kernel conditions.

Contribution

It provides new asymptotic equalities for approximation bounds in $L_p$ metrics for classes of periodic functions with particular Fourier coefficient decay conditions.

Findings

Established asymptotic bounds for approximation errors.

Extended results to classes of $r$-differentiable functions with high smoothness.

Analyzed approximation behavior for functions with specific Fourier coefficient decay.

Abstract

We obtain the asymptotic equalities for the least upper bounds of approximations by interpolation trigonometric polynomials with the equidistant nodes $x_k^{(n-1)}=\frac{2k\pi}{2n-1},\ k\in\mathbb{Z},$ in metrics of the spaces $L_p$ on classes of $2\pi$-periodic functions, representable as convolutions of functions $\varphi, \ \varphi\perp1,$ which belongs to the unit ball of the space $L_1$, and fixed generating kernels in the case where modules of their Fourier coefficients $\psi(k)$ satisfy the condition $\lim\limits_{k\rightarrow\infty} \psi(k+1)/\psi(k)=0.$ We obtain similar estimates on the classes of $r$-differentiable functions $W^r_1$ for the quickly increasing exponents of smoothness $r$ $(r/n\rightarrow\infty)$.