Uniform approximations by Fourier sums on the sets of convolutions of periodic functions of high smoothness - Serdyuk, A.; Stepaniuk, T. (Ukrains’kyi Matematychnyi Zhurnal, 2023)
UDC 517.5On the sets of $2π$-periodic functions $f$ specified by the $(ψ, β)$-integrals of the functions $φ$ from $L_{1},$ we establish Lebesgue-type inequalities in which the uniform norms of deviations of the Fourier sums are expressed via the best approximations by trigonometric polynomials of the functions $φ$ in the mean. It is proved that obtained estimates are asymptotically unimprovable in the case where the sequences $ψ(k)$ approach zero faster than any power function. In some important cases, we establish asymptotic equalities for the exact upper boundaries of the uniform approximations by Fourier sums in the classes of $(ψ, β)$-integrals of the functions $φ$ that belong to the unit ball in the space $L_{1}.$