The Theory of Best Approximation in Normed Linear Spaces

Abstract

It is well known that the problem of best approximation of a function consists in the determination of a function belonging to a fixed family such that its deviation from the given function is a minimum, This problem was first formulated by P. L. Chebyshev, who investigated the approximation of continuous functions by algebraic polynomials of given degree and by rational fractions with numerators and denominators of fixed degree. As a measure of the deviation between two functions, Chebyshev used the maximum of the absolute value of their difference. Subsequently, a number of mathematicians have studied other specialized problems of best approximation whose content is defined by some choice of the measure of deviation and the function set used for the approximation. Among these, we should in the first place note A.A. Markov, Jackson, Bernshtein, de la Vallee-Poussin, Haar, and Kolmogorov. With the development of the theory of normed spaces it became clear that a wide range of problems of best approximation can be put into a general formulation in terms of normed spaces, if the norm of the space is taken as the measure of deviation.