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

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 $r$ $(r/n\rightarrow\infty)$.