# Best Approximation

## Best Approximation

an important concept in the theory of approximations of functions. Let f(x) be an arbitrary continuous function defined on some interval [a, b] and let Φ1(x), Φ2(x), . . .,Φn(x) be a fixed system of continuous functions on this interval. Then the maximum of the expression

(*) ǀf(x) a1Φ1(x) - ... -anΦn(x)

on [a, b] is called the deviation of f(x) from the the polynomial

Pn(x)= a1Φ1(x) + a2Φ2(x) + ... +anΦn(x)

and the minimum of the deviations for all polynomials Pn(x) that is, for all sets of coefficients a1, a2,. . ., an—is called the best approximation of the function f(x) by means of the system Φ1(x), Φ2 (x),..., Φn(x). Denoted by En(f,Φ), the best approximation is the minimum of the maxima, or the minimax. A polynomial P*n(x, f) whose deviation from f(x) is equal to the best approximation (such a polynomial always exists) is said to be the polynomial that deviates least from the function f(x) on the interval [a, b].

The concept of best approximation and of a polynomial that deviates least from the function f(x) were first introduced by P. L. Chebyshev in 1854 in his studies on the theory of mechanisms. It is also possible to consider best approximation when the deviation of a function/(x) from a polynomial Pn(x) is understood to mean, for example, the expression rather than the maximum of the expression (*).

References in periodicals archive ?
The ONS provides data for "employees by jobs" which provides a robust sampling and methodology and the best approximation of employment in the pharmaceutical industry in the UK.
He brings his best approximation of 70's soul and R&B which somehow turns into a modern sounding Beatle-esque outro.
By the estimates (3.6) and (3.7) it follows that the error of the product rule behaves like the best approximation error of the function space f belongs to, except for the extra factor log m.
Only a lawyer can assist the courts in sifting through evidence and the applicable law, and help forge a ruling that is the best approximation of justice duly served."
From these figures, it is clear that the method in  yields the best approximation. The harmonic reconstruction is the worst amongst the 3 methods in this experiment.
As in case of odd p, a user of the algorithms must decide which choice will be more appropriate to select, the best approximation on overall [-1, 1] with worse accuracy on [absolute value of x] < 1, or the best accuracy on the latest one.
 Ivan Singer, Best approximation in normed linear spaces by elements of linear subspaces, Springer-Verlag, 1970.
Section 2 deals with preliminary notions such as the modulus of continuity and the best approximation, but in that use an abstract setting the cosine operator function.
As shown in , the combination of a = 0.339 and b = 5.510, denoted as [Q.sub.a-Borjesson-1] (x), gives the best approximation for x >0, namely:
Let f(t) be arbitrary element of [L.sup.2] [0,1]; since Y is a finite dimensional subspace of [L.sup.2][0,1] space, f(t) has a unique best approximation in Y, say [f.sup.*](t), such that
So, if f is an arbitrary element in H, it has a unique best approximation out of Y such as [y.sub.0]; that is,

Site: Follow: Share:
Open / Close