Difference between revisions of "Bernstein interpolation method"
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| + | 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 | where the | ||
| − | + | $$ | |
| + | T _ {n} (x) = \cos ( n \mathop{\rm arc} \cos x) | ||
| + | $$ | ||
are the [[Chebyshev polynomials|Chebyshev polynomials]]; the | are the [[Chebyshev polynomials|Chebyshev polynomials]]; the | ||
| − | + | $$ | |
| + | x _ {k} ^ {(n)} = \ | ||
| + | \cos \left [ | ||
| + | \frac{(2k-1) \pi }{2n} | ||
| + | \right ] | ||
| + | $$ | ||
are the interpolation nodes; and | are the interpolation nodes; and | ||
| − | + | $$ | |
| + | A _ {k} ^ {(n)} = f (x _ {k} ^ {(n)} ) | ||
| + | $$ | ||
| − | if | + | 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 | + | 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 [[#References|[1]]]. | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> S.N. Bernshtein, , ''Collected works'' , '''2''' , Moscow (1954) pp. 130–140 (In Russian)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> S.N. Bernshtein, , ''Collected works'' , '''2''' , Moscow (1954) pp. 130–140 (In Russian)</TD></TR></table> | ||
| − | |||
| − | |||
====Comments==== | ====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 | + | 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|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 [[#References|[a1]]]. | ||
This method of Bernstein is often used to prove the [[Weierstrass theorem|Weierstrass theorem]] (on approximation). For a generalization of the method (the monotone-operator theorem), see [[#References|[a2]]], Chapt. 3, Sect. 3. See also [[Approximation of functions, linear methods|Approximation of functions, linear methods]]. | This method of Bernstein is often used to prove the [[Weierstrass theorem|Weierstrass theorem]] (on approximation). For a generalization of the method (the monotone-operator theorem), see [[#References|[a2]]], Chapt. 3, Sect. 3. See also [[Approximation of functions, linear methods|Approximation of functions, linear methods]]. | ||
Latest revision as of 10:58, 29 May 2020
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 |
How to Cite This Entry:
Bernstein interpolation method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bernstein_interpolation_method&oldid=11602
Bernstein interpolation method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bernstein_interpolation_method&oldid=11602
This article was adapted from an original article by P.P. Korovkin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article