Uniform approximations by Fourier sums on the sets of convolutions of periodic functions of high smoothness

Abstract: On the sets of $2\pi$-periodic functions $f$, which are defined with a help of $(\psi, \beta)$-integrals of the functions $\varphi$ from $L_{1}$, we establish Lebesgue-type inequalities, in which the uniform norms of deviations of Fourier sums are expressed via the best approximations by trigonometric polynomials of the functions $\varphi$. We prove that obtained estimates are best possible, in the case when the sequences $\psi(k)$ decrease to zero faster than any power function. In some important cases we establish the asymptotic equalities for the exact upper boundaries of uniform approximations by Fourier sums on the classes of $(\psi, \beta)$-integrals of the functions $\varphi$, which belong to the unit ball of the space $L_{1}$.