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]).

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)