Approximation of ψ - integrals−integrals of periodic functions by Fourier sums (small smoothness). Iof periodic functions by Fourier sums (small smoothness). I - Stepanets, O. I. (Ukrains’kyi Matematychnyi Zhurnal, 1998)
We investigate the rate of convergence of Fourier series on the classes L^ψ N in the uniform and integral metrics. The results obtained are extended to the case where the classes L^ψ N are the classes of convolutions of functions from N 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^ψ N which are solutions of the Kolmogorov-Nikol’skii problem. In addition, we establish an analog of the well-known Lebesgue inequality.