METHODOLOGICAL ASPECTS OF THE STUDY OF THE THEORY OF APPROXIMATION OF A FUNCTION BY ORTHOGONAL POLYNOMIALS

Abstract

. In the presented article, the general theory and some significant properties of continuous approximation are presented, and cases of approximation by orthogonal polynomials are considered. There are a number of areas where orthogonal polynomials are widely used. Traditional applications include orthogonal expansions, least squares approximation, interpolation, and numerical quadrature. The article generalizes and systematizes knowledge in the field of approximation by orthogonal polynomials. Orthogonal polynomials are useful in many areas of numerical analysis and are useful for function approximation, numerical integration, and the numerical solution of differential and integral equations. Traditional applications of orthogonal polynomials include orthogonal expansions, least squares approximation, interpolation, and numerical quadrature. Somewhat less well known are the connections of orthogonal polynomials with the Pad approximation, when the underlying power series has moments as coefficients. More recent applications include probability theories, e.g., birth and death processes, Carlin and McGregor, coding theory, Sloan, Delsarte, scattering theory, Case, relaxation methods in numerical linear algebra, Stiefel, as well as in prediction theory and Toeplitz matrix inversion, where Sego polynomials on the unit the circles are noticeably involved, Kailat. Sego’s theorem on the asymptotic behavior of Toeplitz determinants also has important applications in statistical mechanics, McCoy.