On the approximation of the | sin |s x function by rational trigonometric operators of the Fejér type

Approximation by trigonometric Fourier series is a well-developed branch of the theory of approximation by polynomials. Methods of approximation by rational trigonometric Fourier series have not been researched so deeply yet. In particular, rational trigonometric operators of the Fejér type have not been used in the rational approximation with free poles. In this paper, we consider the approximation of the function | sin | , (0;2), ∈ s x s by rational trigonometric operators of the Fejér type. An integral representation of the remainder for the above-mentioned approximation is obtained. An estimate of approximations is found in the points of analyticity of the function | sin |s x under the condition that the corresponding system of rational functions is complete. It is shown that the order of uniform approximation in the case of approximation by rational Fejér functions with two geometrically different poles is higher than the order of approximation by trigonometric polynomials. As a result, an asymptotic estimation of the uniform approximation by trigonometric Fejér sums in the polynomial case is obtained.