Rate of convergence of Fourier series on the classes of $\bar \Psi$-integrals-integrals

Abstract

We introduce the notion of $\bar \Psi$-integrals of 2π-periodic summable functions f, f ε L, on the basis of which the space L is decomposed into subsets (classes) $L^{\bar \Psi }$. We obtain integral representations of deviations of the trigonometric polynomials U _n(f;x;Λ) generated by a given Λ-method for summing the Fourier series of functions $f{\text{ }}\varepsilon {\text{ }}L^{\bar \Psi }$. On the basis of these representations, the rate of convergence of the Fourier series is studied for functions belonging to the sets $L^{\bar \Psi }$ in uniform and integral metrics. Within the framework of this approach, we find, in particular, asymptotic equalities for upper bounds of deviations of the Fourier sums on the sets $L^{\bar \Psi }$, which give solutions of the Kolmogorov-Nikol'skii problem. We also obtain an analog of the well-known Lebesgue inequality.