P.G. Potseiko, E.A. Rovba. A Fejér rational integral operator on a closed interval and approximation of functions with a power-law singularity ... P. 170-189

Rational approximations of continuous functions and functions with a power-law singularity on a closed interval are studied by means of integral Fejér-type operators. Upper estimates of approximations of continuous functions on a closed interval are derived; the estimates are expressed in terms of the modulus of continuity and depend on the position of a point in the interval. Rational approximations of the function $(1-x)^\gamma$, $\gamma\in (0,1)$, on the interval $[-1,1]$ are studied. Upper estimates of uniform approximations in terms of the corresponding majorant and an asymptotic expression as $n\to\infty$ of this majorant are found. In the case of a fixed number of poles of the approximating function, optimal values of the parameters are obtained, for which the majorant of the uniform approximations decreases at the highest rate. A consequence of the results obtained is asymptotic estimates of approximations of some specific functions by Fejér sums of polynomial Fourier-Chebyshev series.

Keywords: rational approximations, Fejér integral operator, pointwise and uniform estimates of approximations, modulus of continuity, function with a power-law singularity, asymptotic estimates

Received May 15, 2023

Revised December 18, 2023

Accepted December 25, 2023

Funding Agency: This work was supported by the National Program for Scientific Research of the Republic of Belarus “Convergence 2020” (project no. 20162269).

Pavel G. Potseiko, Ph.D., Faculty of Mathematics and Informatics Yanka Kupala State University of Grodno (Belarus) Ozheshko St., 22, 230023, Grodno, Belarus; e-mail: pahamatby@gmail.com

Yevgeniy A. Rovba, Dr. Phys.-Math. Sci., Prof., Faculty of Mathematics and Informatics Yanka Kupala State University of Grodno (Belarus) Ozheshko St., 22, 230023, Grodno, Belarus; e-mail: rovba.ea@gmail.com

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Cite this article as: P.G. Potseiko, E.A. Rovba. A Fejér rational integral operator on a closed interval and approximation of functions with a power-law singularity. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2024, vol. 30, no. 1, pp. 170–189.