Characterization Of Best Approximations Research Articles

Introduction to Approximation Theory

Introduction: 1 Examples and prospectus 2 Metric spaces 3 Normed linear spaces 4 Inner-product spaces 5 Convexity 6 Existence and unicity of best approximations 7 Convex functions The Tchebycheff Solution of Inconsistent Linear Equations: 1 Introduction 2 Systems of equations with one unknown 3 Characterization of the solution 4 The special case 5 Polya's algorithm 6 The ascent algorithm 7 The descent algorithm 8 Convex programming Tchebycheff Approximation by Polynomials and Other Linear Families: 1 Introduction 2 Interpolation 3 The Weierstrass theorem 4 General linear families 5 the unicity problem 6 Discretization errors: General theory 7 Discretization: Algebraic polynomials. The inequalities of Markoff and Bernstein 8 Algorithms Least-squares Approximation and Related Topics: 1 Introduction 2 Orthogonal systems of polynomials 3 Convergence of orthogonal expansions 4 Approximation by series of Tchebycheff polynomials 5 Discrete least-squares approximation 6 The Jackson theorems Rational Approximation: 1 Introduction 2 Existence of best rational approximations 3 The characterization of best approximations 4 Unicity Continuity of best-approximation operators 5 Algorithms 6 Pade Approximation and its generalizations 7 Continued fractions Some Additional Topics: 1 The Stone approximation theorem 2 The Muntz theorem 3 The converses of the Jackson theorems 4 Polygonal approximation and bases in $C[a, b]$ 5 The Kharshiladze-Lozinski theorems 6 Approximation in the mean Notes References Index.

Applications of the Hahn-Banach Theorem in Approximation Theory

Applications of the Hahn-Banach Theorem in Approximation Theory

Chebyshev Approximation by Exponentials

are considered. For n = 1 the theory is given in [3]. The principal theoretical questions of existence, uniqueness and characterization of best approximations are answered for this approximating function. The problem of the computation of best approximations is unsolved. This problem is complicated by three factors: the transcendental nature of the functions involved; the variability of the characterizing property of best approximations and the possible nonexistence of best approximations on a finite point set. In order to obtain a unified theory the approximating function E(A, x) actually considered is the set of all functions of the form