Abstract
Introducing some classes of the functions defined by Poisson integrals on the segment \( [-1,1] \) and studying approximations by Fourier–Chebyshev rational integral operators on the classes, we establish integral expressions for approximations and upper bounds for uniform approximations. In the case of boundary functions with a power singularity on \( [-1,1] \), we find the upper bounds for pointwise and uniform approximations and an asymptotic expression for a majorant of uniform approximations in terms of rational functions with a fixed number of prescribed geometrically distinct poles. Considering two geometrically distinct poles of the approximant of even multiplicity, we obtain asymptotic estimates for the best uniform approximation by this method with a higher convergence rate than polynomial analogs.
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The authors were supported by the Fundamental Research Program “Convergence 2020” (Grant 20162269).
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Translated from Sibirskii Matematicheskii Zhurnal, 2021, Vol. 62, No. 2, pp. 362–386. https://doi.org/10.33048/smzh.2021.62.209
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Potseiko, P.G., Rovba, E.A. Approximations on Classes of Poisson Integrals by Fourier–Chebyshev Rational Integral Operators. Sib Math J 62, 292–312 (2021). https://doi.org/10.1134/S0037446621020099
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DOI: https://doi.org/10.1134/S0037446621020099
Keywords
- class of Poisson integrals
- rational integral operators
- Fourier series
- pointwise and uniform approximation
- asymptotic estimates
- accurate constants