Approximation of \bar \psi - integrals of periodic functions by Fourier sums (small smoothness). Iof periodic functions by Fourier sums (small smoothness). I

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 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^{\bar \psi } $ N which are solutions of the Kolmogorov-Nikol’skii problem. In addition, we establish an analog of the well-known Lebesgue inequality.