Approximation Methods for Solutions of Differential and Integral Equations

Part 1 Introduction to approximation theory: Chebyshev theory of approximation of real functions - existence theorem; Chebyshev theorem on the characterization of a polynomial of best uniform approximation; uniqueness theorem; lower estimate of the value of best approximation; uniqueness Chebyshev polynomials; Chebyshev polynomials of the second kind; an efficient construction of polynomials least deviating from zero with polynomial weights; the notion of snakes; Chebyshev approximation of functions by rational polynomials; linear methods of Fourier series summation and approximation of continuous functions - linear methods of Fourier series summation; discrete polynomial linear operators; Lagrange interpolation polynomials - fundamental Lagrange polynomials of a general form; Lebesgue functions and constants; periodic Lagrange polynomials constructed with equidistant nodes and corresponding algebraic Lagrange polynomials; explicit formulas for Lebesgue constants in interpolation constructed with optimal nodes; the representation of Lebesgue constants as sums of asymptotic series; estimating the best approximation of analytic functions on a segment by polynomials - basic definitions; Bernstein theorem; estimates for the norms of trigonometric and algebraic polynomials by their values at a discrete system of points. Part 2 A simple approach to Fredholm integral equations of the second kind and application of linear methods for approximating their solutions. Part 3 Iteration method for approximately solving differential and integral equations. Part 4 Approximation method for solving ordinary linear differential equations. Part 5 Approximation of solutions of partial differential equations. Part 6 Theory of transcendental functions and their approximation. Part 7 Application of the a-method to the rational approximation and integral representation of functions. (Part contents)