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 [1].

References

[1]	S.N. Bernshtein, , Collected works , 2 , Moscow (1954) pp. 130–140 (In Russian)

Comments

This method of interpolation seems not very well known in the West. There is, however, a well-known method of Bernstein that uses the special interpolation nodes $k/n$, $k = 0 \dots n$, for bounded functions on $[0, 1]$. This method is given by the Bernstein polynomials. The sequence of Bernstein polynomials $B _ {n} (f; x)$ constructed for a bounded function $f$ on $[0, 1]$ converges to $f (x)$ at each point of continuity $x \in [0, 1]$ of $f$. If $f$ is continuous on $[0, 1]$, the sequence converges uniformly (to $f$) on $[0, 1]$. If $f$ is differentiable, $B _ {n} ^ { \prime } (f; x) \rightarrow f ^ { \prime } (x)$( at each point of continuity of $f ^ { \prime }$), cf [a1].

This method of Bernstein is often used to prove the Weierstrass theorem (on approximation). For a generalization of the method (the monotone-operator theorem), see [a2], Chapt. 3, Sect. 3. See also Approximation of functions, linear methods.

References

[a1]	P.J. Davis, "Interpolation and approximation" , Dover, reprint (1975)
[a2]	E.W. Cheney, "Introduction to approximation theory" , Chelsea, reprint (1982) pp. 203ff