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Ukrainian Mathematical Journal

, Volume 49, Issue 8, pp 1201–1251 | Cite as

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

  • A. I. Stepanets
Article

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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Plenum Publishing Corporation 1998

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  • A. I. Stepanets

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