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), . . .,Φ_n(x) be a fixed system of continuous functions on this interval. Then the maximum of the expression

(*) ǀf(x) a₁Φ₁(x) - ... -a_nΦ_n(x)

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

P_n(x)= a₁Φ₁(x) + a₂Φ₂(x) + ... +a_nΦ_n(x)

and the minimum of the deviations for all polynomials P_n(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),..., Φ_n(x). Denoted by E_n(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 P_n(x) is understood to mean, for example, the expression

rather than the maximum of the expression (*).