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We find two-sides estimates for the best uniform approximations of classes of convolutions of $2\pi$-periodic functions from unit ball of the space $L_p, 1 \le p <\infty,$ with fixed kernels, modules of Fourier coefficients of which 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.

Asymptotic Estimates for the Best Uniform Approximations of Classes of Convolution of Periodic Functions of High Smoothness