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