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 (*).