Fejer means of rational Fourier – Chebyshev series and approximation of function $|x|^{s}$

P. G. Potseiko ; Y. A. Rovba

Journal of the Belarusian State University. Mathematics and Informatics, Tome 3 (2019), pp. 18-34

Résumé

Approximation properties of Fejer means of Fourier series by Chebyshev – Markov system of algebraic fractions and approximation by Fejer means of function $|x|^{s}, 0$, on the interval $[-1,1]$, are studied. One orthogonal system of Chebyshev – Markov algebraic fractions is considers, and Fejer means of the corresponding rational Fourier – Chebyshev series is introduce. The order of approximations of the sequence of Fejer means of continuous functions on a segment in terms of the continuity module and sufficient conditions on the parameter providing uniform convergence are established. A estimates of the pointwise and uniform approximation of the function $|x|^{s}, 0$, on the interval $[-1,1]$ , the asymptotic expressions under $n\rightarrow \infty$ of majorant of uniform approximations, and the optimal value of the parameter, which provides the highest rate of approximation of the studied functions are sums of rational use of Fourier – Chebyshev are found.

Détail

Keywords: Fourier – Chebyshev series; partial sums; Fejer means; modulus of continuity; uniform convergence; asymptotic estimates; exact constants.