Polynomial of best approximation

A polynomial furnishing the best approximation of a function in some metric, relative to all polynomials constructed from a given (finite) system of functions. If is a normed linear function space (such as or , ), and if

is a system of linearly independent functions in , then for any the (generalized) polynomial of best approximation

(*)

defined by the relation

exists. The polynomial of best approximation is unique for all if is a space with a strictly convex norm (i.e. if and , then ). This is the case for , . In , which has a norm that is not strictly convex, the polynomial of best approximation for any is unique if is a Chebyshev system on , i.e. if each polynomial

has at most zeros on . In particular, one has uniqueness in the case of the (usual) algebraic polynomials in , and also for the trigonometric polynomials in the space of continuous -periodic functions on the real line, with the uniform metric. If the polynomial of best approximation exists and is unique for any , it is a continuous function of .

Necessary and sufficient conditions for a polynomial to be a best approximation in the spaces and are known. For example, one has Chebyshev's theorem: If is a Chebyshev system, then the polynomial (*) is a polynomial of best approximation for a function in the metric of if and only if there exists a system of points , , at which the difference

assumes values

and, moreover,

The polynomial (*) is a polynomial of best approximation for a function , , in the metric of that space, if and only if for ,

In , the conditions

are sufficient for to be a polynomial of best approximation for , and if the measure of the set of all points at which is zero, they are also necessary; see also Markov criterion.

There exist algorithms for the approximate construction of polynomials of best uniform approximation (see e.g. [3], [5]).

References

[1]	N.I. [N.I. Akhiezer] Achiezer, "Theory of approximation" , F. Ungar (1956) (Translated from Russian)
[2]	N.P. Korneichuk, "Extremal problems in approximation theory" , Moscow (1976) (In Russian)
[3]	V.K. Dzyadyk, "Introduction to the theory of uniform approximation of functions by polynomials" , Moscow (1977) (In Russian)
[4]	V.M. Tikhomirov, "Some problems in approximation theory" , Moscow (1976) (In Russian)
[5]	P.J. Laurent, "Approximation et optimisation" , Hermann (1972)
[6]	E.Ya. Remez, "Foundations of numerical methods of Chebyshev approximation" , Kiev (1969) (In Russian)

References

[a1]	A.M. Pinkus, "On -approximation" , Cambridge Univ. Press (1989)
[a2]	A. Schönhage, "Approximationstheorie" , de Gruyter (1971)
[a3]	G.A. Watson, "Approximation theory and numerical methods" , Wiley (1980)
[a4]	E.W. Cheney, "Introduction to approximation theory" , McGraw-Hill (1966) pp. Chapts. 4&6
[a5]	M.J.D. Powell, "Approximation theory and methods" , Cambridge Univ. Press (1981)
[a6]	J.R. Rice, "The approximation of functions" , 1. Linear theory , Addison-Wesley (1964)
[a7]	T.J. Rivlin, "An introduction to the approximation of functions" , Blaisdell (1969)