Approximation of $\bar \psi - Integrals$ of periodic functions by Fourier sums (small smoothness). IIof periodic functions by Fourier sums (small smoothness). II

Abstract

We investigate the rate of convergence of Fourier series on the classes $L^{\bar \psi }$ N in the uniform and integral metrics. The results obtained are extended to the case where the classes $L^{\bar \psi }$ N are the classes of convolutions of functions from ℜ with kernels with slowly decreasing coefficients. In particular, we obtain asymptotic equalities for the upper bounds of deviations of the Fourier sums on the sets $L^{\bar \psi }$ N, which are solutions of the Kolmogorov-Nikol’skii problem. In addition, we establish an analog of the well-known Lebesgue inequality.