Vallée Poussin sums of rational Fourier–Chebyshev integral operators and approximations of the Markov function

Rational approximations of the Markov function on the segment <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket negative 1 comma 1 right-bracket"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">[-1,1]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are studied. The Vallée Poussin sums of rational integral Fourier–Chebyshev operators as an approximation apparatus with a fixed number of geometrically distinct poles are chosen. For the constructed rational approximation method, integral representations of approximations and upper estimates of uniform approximations are established. For the Markov function with a measure whose derivative is a function that has a power-law singularity on the segment <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket negative 1 comma 1 right-bracket"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">[-1, 1]</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, upper estimates of pointwise and uniform approximations and an asymptotic expression of a majorant of uniform approximations are found. The values of the parameters of the approximating function are determined at which the best uniform rational approximations are provided by this method. It is shown that in this case they have a higher rate of decrease in comparison with the corresponding polynomial analogs. As a corollary, rational approximations on a segment by Vallée Poussin sums of some elementary functions representable by a Markov function are considered.