Asymptotic estimates for the best uniform approximations of classes of convolution of periodic functions of high smoothness

Abstract

We find two-sided estimates for the best uniform approximations of classes of convolutions of $2\pi$-periodic functions from a unit ball of the space $L_p, 1 \le p <\infty,$ with fixed kernels such that the moduli of their Fourier coefficients satisfy the condition $\sum\limits_{k=n+1}^\infty\psi(k)<\psi(n).$ In the case of $\sum\limits_{k=n+1}^\infty\psi(k)=o(1)\psi(n),$ the obtained estimates become the asymptotic equalities.