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 $\phi ,\phi \perp 1,$ 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 $\underset{k\to \infty }{\lim }\psi (k+1)/\psi (k)=0.$ We obtain similar estimates on the classes of $r$-differentiable functions $W_1^r$ for the quickly increasing exponents of smoothness $r$ $(r/n\to \infty )$.

8 pages, in Russian