Approximations of one singular integral on an interval by Fourier--Chebyshev rational integral operators

P. G. Potseiko ; E. A. Rovba

Sbornik. Mathematics, Tome 215 (2024) no. 7, pp. 953-992

Résumé

We study approximations on the interval $[-1,1]$ of singular integrals of the form $$ \widehat{f}(x)=\int_{-1}^{1}\frac{f(t)}{t-x}\sqrt{1-t^2}\,dt, \qquad x \in [-1,1], $$ by two rational integral operators related to each other in a certain sense. The first is the Fourier–Chebyshev integral operator associated with the Chebyshev–Markov system of rational functions. The second operator is its image under the transformation by the singular integral under consideration. Approximative properties of the corresponding polynomial analogues of both operators are studied in the case where the density of the singular integral satisfies a Hölder condition of exponent $\alpha \in (0,1]$ on $[-1,1]$. Rational approximations on $[-1,1]$ of the singular integral with power-law singular density are investigated. In the two cases under consideration the approximating rational functions have arbitrary many fixed geometrically different poles or the parameters of the approximating rational functions are modifications of the ‘Newman’ parameters. Bibliography: 34 titles.

Détail

Keywords: singular integral on an interval, Fourier–Chebyshev rational integral operators, uniform estimate, Laplace method, strong asymptotics.