Best approximation

of a function by functions from a fixed set

The quantity

where is the error of approximation (see Approximation of functions, measure of). The concept of a best approximation is meaningful in an arbitrary metric space when is defined by the distance between and ; in this case is the distance from to the set . If is a normed linear space, then for a fixed the best approximation

(1)

may be regarded as a functional defined on (the functional of best approximation).

The functional of best approximation is continuous, whatever the set . If is a subspace, the functional of best approximation is a semi-norm, i.e.

and

for any . If is a finite-dimensional subspace, then for any there exists an element (an element of best approximation) at which the infimum in (1) is attained:

In a space with a strictly convex norm, the element of best approximation is unique.

Through the use of duality theorems, the best approximation in a normed linear space can be expressed in terms of the supremum of the values of certain functionals from the adjoint space (see, e.g. [5], [8]). If is a closed convex subset of , then for any

(2)

in particular, if is a subspace, then

(3)

where is the set of functionals in such that for any . In the function spaces or , the right-hand sides of (2) and (3) take explicit forms depending on the form of the linear functional. In a Hilbert space , the best approximation of an element by an -dimensional subspace is obtained by orthogonal projection on and can be calculated; one has:

where form a basis of and is the Gram determinant, the elements of which are the scalar products , . If is an orthonormal basis, then

In the space one has the following estimate for the best uniform approximation of a function by an -dimensional Chebyshev subspace (the de la Vallée-Poussin theorem): If for some function there exist points , , for which the difference

takes values with alternating signs, then

For best approximations in see Markov criterion. In several important cases, the best approximations of functions by finite-dimensional subspaces can be bounded from above in terms of differential-difference characteristics (e.g. the modulus of continuity) of the approximated function or its derivatives.

The concept of a best uniform approximation of continuous functions by polynomials is due to P.L. Chebyshev (1854), who developed the theoretical foundations of the concept and established a criterion for polynomials of best approximation in the metric space (see Polynomial of best approximation).

The best approximation of a class of functions is the supremum of the best approximations of the functions in the given class by a fixed set of functions , i.e. the quantity

The number characterizes the maximum deviation (in the specific metric chosen) of the class from the approximating set and indicates the minimal possible error to be expected when approximating an arbitrary function by functions of .

Let be a subset of a normed linear function space , let be a linearly independent system of functions in and let , be the subspaces generated by the first elements of this system. By investigating the sequence , one can draw conclusions regarding both the structural and smoothness properties of the functions in and the approximation properties of the system relative to . If is a Banach function space and is closed in , i.e. , then as if and only if is a compact subset of .

In various important cases, e.g. when the are subspaces of trigonometric polynomials or periodic splines, and the class is defined by conditions imposed on the norm or on the modulus of continuity of some derivative , the numbers can be calculated explicitly [5]. In the non-periodic case, results are available concerning the asymptotic behaviour of as .

References

[1]	P.L. Chebyshev, "Complete collected works" , 2 , Moscow (1947) (In Russian)
[2]	N.I. [N.I. Akhiezer] Achiezer, "Theory of approximation" , F. Ungar (1956) (Translated from Russian)
[3]	V.K. Dzyadyk, "Introduction to the theory of uniform approximation of functions by polynomials" , Moscow (1977) (In Russian)
[4]	V.L. Goncharov, "The theory of interpolation and approximation of functions" , Moscow (1954) (In Russian)
[5]	N.P. Korneichuk, "Extremal problems in approximation theory" , Moscow (1976) (In Russian)
[6]	S.M. Nikol'skii, "Approximation of functions of several variables and imbedding theorems" , Springer (1975) (Translated from Russian)
[7]	A.F. Timan, "Theory of approximation of functions of a real variable" , Pergamon (1963) (Translated from Russian)
[8]	V.M. Tikhomirov, "Some problems in approximation theory" , Moscow (1976) (In Russian)
[9]	P.J. Laurent, "Approximation et optimisation" , Hermann (1972)

Comments

In Western literature an element, a functional or a polynomial of best approximation is also called a best approximation.

References

[a1]	G.G. Lorentz, "Approximation of functions" , Holt, Rinehart & Winston (1966)
[a2]	E.W. Cheney, "Introduction to approximation theory" , Chelsea, reprint (1982) pp. 203ff
[a3]	J.R. Rice, "The approximation of functions" , 1. Linear theory , Addison-Wesley (1964)
[a4]	A. Pinkus, "-widths in approximation theory" , Springer (1985) (Translated from Russian)