Polynomial of best approximation
A polynomial furnishing the best approximation of a function $ x ( t) $ in some metric, relative to all polynomials constructed from a given (finite) system of functions. If $ X $ is a normed linear function space (such as $ C [ a, b] $ or $ L _ {p} ( a, b) $, $ p \geq 1 $), and if
$$ U _ {n} = \{ u _ {1} ( t) \dots u _ {n} ( t) \} $$
is a system of linearly independent functions in $ X $, then for any $ x \in X $ the (generalized) polynomial of best approximation
$$ \tag{* } \widetilde{u} ( t) = \widetilde{u} ( x, t) = \ \sum _ {k = 1 } ^ { n } \widetilde{c} _ {k} u _ {k} ( t), $$
defined by the relation
$$ \| x - \widetilde{u} \| = \ \min _ {\{ c _ {k} \} } \left \| x - \sum _ {k = 1 } ^ { n } c _ {k} u _ {k} \right \| , $$
exists. The polynomial of best approximation is unique for all $ x \in X $ if $ X $ is a space with a strictly convex norm (i.e. if $ \| x \| = \| y \| $ and $ x \neq y $, then $ \| x + y \| < \| x \| + \| y \| $). This is the case for $ L _ {p} ( a, b) $, $ 1 < p < \infty $. In $ C [ a, b] $, which has a norm that is not strictly convex, the polynomial of best approximation for any $ x \in C [ a, b] $ is unique if $ U _ {n} $ is a Chebyshev system on $ [ a, b] $, i.e. if each polynomial
$$ u ( t) = \sum _ {k = 1 } ^ { n } c _ {k} u _ {k} ( t) \neq 0 $$
has at most $ n - 1 $ zeros on $ [ a, b] $. In particular, one has uniqueness in the case of the (usual) algebraic polynomials in $ C [ a, b] $, and also for the trigonometric polynomials in the space $ C _ {2 \pi } $ of continuous $ 2 \pi $- periodic functions on the real line, with the uniform metric. If the polynomial of best approximation exists and is unique for any $ x \in X $, it is a continuous function of $ x $.
Necessary and sufficient conditions for a polynomial to be a best approximation in the spaces $ C[ a, b] $ and $ L _ {p} [ a, b] $ are known. For example, one has Chebyshev's theorem: If $ U _ {n} $ is a Chebyshev system, then the polynomial (*) is a polynomial of best approximation for a function $ x \in C [ a, b] $ in the metric of $ C [ a, b] $ if and only if there exists a system of $ n + 1 $ points $ t _ {i} $, $ a \leq t _ {1} < {} \dots < t _ {n + 1 } \leq b $, at which the difference
$$ \Delta ( t) = x ( t) - \widetilde{u} ( t) $$
assumes values
$$ \pm \max _ {a \leq t \leq b } | \Delta ( t) | $$
and, moreover,
$$ \Delta ( t _ {i + 1 } ) = - \Delta ( t _ {i} ),\ \ i = 1 \dots n. $$
The polynomial (*) is a polynomial of best approximation for a function $ x ( t) \in L _ {p} [ a, b] $, $ p > 1 $, in the metric of that space, if and only if for $ k = 1 \dots n $,
$$ \int\limits _ { a } ^ { b } u _ {k} ( t) | x ( t) - \widetilde{u} ( t) | ^ {p - 1 } \ \mathop{\rm sign} [ x ( t) - \widetilde{u} ( t)] dt = 0. $$
In $ L _ {1} [ a, b] $, the conditions
$$ \int\limits _ { a } ^ { b } u _ {k} ( t) \mathop{\rm sign} [ x ( t) - \widetilde{u} ( t)] dt = 0,\ \ k = 1 \dots n, $$
are sufficient for $ \widetilde{u} ( t) $ to be a polynomial of best approximation for $ x \in L _ {1} [ a, b] $, and if the measure of the set of all points $ t \in ( a, b) $ at which $ x ( t) = \widetilde{u} ( t) $ 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]).
[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)
[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)
Polynomial of best approximation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Polynomial_of_best_approximation&oldid=48237
This article was adapted from an original article by N.P. KorneichukV.P. Motornyi (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
Retrieved from "https://encyclopediaofmath.org/index.php?title=Polynomial_of_best_approximation&oldid=48237"