# Rate of convergence of Fourier series on the classes of \(\bar \Psi \)-integrals-integrals

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## 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.

## Keywords

Fourier Series Entire Function Fourier Coefficient Trigonometric Polynomial Summable Function## Preview

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