Rate of convergence of Fourier series on the classes of ψ-integrals
We introduce the notion of ψ-integrals of 2π-periodic summable functions f, f ε L, on the basis of which the space L is decomposed into subsets (classes) L^ψ . 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 ε L^ψ . On the basis of these representations, the rate of convergence of the Fourier series is studied for functions belonging to the sets L^ψ 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^ψ , which give solutions of the Kolmogorov-Nikol’skii problem. We also obtain an analog of the well-known Lebesgue inequality., number=8, journal=Ukrains’kyi Matematychnyi Zhurnal, author=Stepanets, O. I., year=1997, month=Aug., pages=1069–1113