# Algebraic polynomial of best approximation

A polynomial deviating least from a given function. More precisely, let a measurable function $f(x)$ be in $L _ {p} [a, b]$( $p \geq 1$) and let $H _ {n}$ be the set of algebraic polynomials of degree not exceeding $n$. The quantity
$$\tag{* } E _ {n} (f) _ {p} = \inf _ {P _ {n} (x) \in H _ {n} } \| f (x) - P _ {n} (x) \| _ {L _ {p} [ a , b ] }$$
is called the best approximation, while a polynomial for which the infimum is attained is known as an algebraic polynomial of best approximation in $L _ {p} [a, b]$. Polynomials which deviate least from a given continuous function in the uniform metric ( $p = \infty$) were first encountered in the studies of P.L. Chebyshev (1852), who continued to study them in 1856 . The existence of algebraic polynomials of best approximation was established by E. Borel . Chebyshev proved that $P _ {n} ^ {0} (x)$ is an algebraic polynomial of best approximation in the uniform metric if and only if Chebyshev alternation occurs in the difference $f(x) - P _ {n} ^ {0} (x)$; in this case such a polynomial is unique. If $p > 1$, the algebraic polynomial of best approximation is unique due to the strict convexity of the space $L _ {p}$. If $p = 1$, it is not unique, but it has been shown by D. Jackson  to be unique for continuous functions. The rate of convergence of $E _ {n} {(f) } _ {p}$ to zero is given by Jackson's theorems (cf. Jackson theorem).
In a manner similar to (*) an algebraic polynomial of best approximation is defined for functions in a large number of unknowns, say $m$. If the number of variables $m \geq 2$, an algebraic polynomial of best approximation in the uniform metric is, in general, not unique.