# Bernstein interpolation method

A sequence of algebraic polynomials converging uniformly on $[-1, 1]$ to a function $f(x)$ that is continuous on this interval. More precisely, Bernstein's interpolation method is a sequence of algebraic polynomials

$$P _ {n} (f; x) = \ \frac{\sum _ { k=1 } ^ { n } A _ {k} ^ {(n)} T _ {n} (x) }{T _ {n} (x _ {k} ^ {(n)} )(x-x _ {k} ^ {(n)} ) } , \ n = 1, 2 \dots$$

where the

$$T _ {n} (x) = \cos ( n \mathop{\rm arc} \cos x)$$

are the Chebyshev polynomials; the

$$x _ {k} ^ {(n)} = \ \cos \left [ \frac{(2k-1) \pi }{2n} \right ]$$

are the interpolation nodes; and

$$A _ {k} ^ {(n)} = f (x _ {k} ^ {(n)} )$$

if $k \neq 2ls, l$ is an arbitrary positive integer, $n = 2 l q + r$, $q \geq 1$, $0 \leq r < 2l$, $s = 1 \dots q;$ otherwise

$$A _ {2ls} ^ {(n)} = \ \sum _ { i=0 } ^ { l-1 } f(x _ {2l (s - 1) + 2i + 1 } ^ {(n)} ) - \sum _ { i=1 } ^ { l-1 } f (x _ {2l (s - 1) + 2i } ^ {(n)} ).$$

The ratio between the degree of the polynomial $P _ {n} (f; x)$ and the number of points at which $P _ {n} (f; x)$ equals $f(x)$ is $(n - 1)/(n - q)$, which tends to $2l/(2l - 1)$ as $n \rightarrow \infty$; if $l$ is sufficiently large, this limit is arbitrary close to one. The method was introduced by S.N. Bernstein [S.N. Bernshtein] in 1931 .

How to Cite This Entry:
Bernstein interpolation method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bernstein_interpolation_method&oldid=46026
This article was adapted from an original article by P.P. Korovkin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article