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Algebraic polynomials defined by the formula
 
Algebraic polynomials defined by the formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015730/b0157301.png" /></td> </tr></table>
+
$$
 +
B _ {n} (f; x )  = \
 +
B _ {n} (x ) =
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015730/b0157302.png" /></td> </tr></table>
+
$$
 +
= \
 +
\sum _ { k=0 } ^ { n }  f \left ( {
 +
\frac{k}{n}
 +
} \right ) \left (
 +
\begin{array}{c}
 +
n \\
 +
k
 +
\end{array}
 +
\right ) x  ^ {k} (1-x)  ^ {n-k} ,\  n = 1, 2 ,\dots .
 +
$$
  
Introduced by S.N. Bernshtein in 1912 (cf. ). The sequence of Bernstein polynomials converges uniformly to a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015730/b0157303.png" /> on the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015730/b0157304.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015730/b0157305.png" /> is continuous on this segment. For a function which is bounded by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015730/b0157306.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015730/b0157307.png" />, with a discontinuity of the first kind,
+
Introduced by S.N. Bernshtein in 1912 (cf. ). The sequence of Bernstein polynomials converges uniformly to a function $  f $
 +
on the segment 0 \leq  x \leq  1 $
 +
if $  f $
 +
is continuous on this segment. For a function which is bounded by $  C $,
 +
$  0 < C < 1 $,  
 +
with a discontinuity of the first kind,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015730/b0157308.png" /></td> </tr></table>
+
$$
 +
B _ {n} (f; C)  \rightarrow \
 +
{
 +
\frac{f (C _ {-} ) + f (C _ {+} ) }{2}
 +
} .
 +
$$
  
 
The equation
 
The equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015730/b0157309.png" /></td> </tr></table>
+
$$
 +
B _ {n} (f; c) - f (c)  = \
 +
 
 +
\frac{f ^ { \prime\prime } (c)c(1-c) }{2n}
 +
 
 +
+ o \left (
 +
\frac{1}{n}
 +
\right )
 +
$$
  
is valid if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015730/b01573010.png" /> is twice differentiable at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015730/b01573011.png" />. If the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015730/b01573012.png" />-th derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015730/b01573013.png" /> of the function is continuous on the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015730/b01573014.png" />, the convergence
+
is valid if $  f $
 +
is twice differentiable at the point $  c $.  
 +
If the $  k $-
 +
th derivative $  f ^ { (k) } $
 +
of the function is continuous on the segment 0 \leq  x \leq  1 $,  
 +
the convergence
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015730/b01573015.png" /></td> </tr></table>
+
$$
 +
B _ {n} ^ { (k) } (f; x)  \rightarrow  f ^ { (k) } (x)
 +
$$
  
is uniform on this segment. A study was made ([[#References|[1b]]], [[#References|[5]]]) of the convergence of Bernstein polynomials in the complex plane if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015730/b01573016.png" /> is analytic on the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015730/b01573017.png" />.
+
is uniform on this segment. A study was made ([[#References|[1b]]], [[#References|[5]]]) of the convergence of Bernstein polynomials in the complex plane if $  f $
 +
is analytic on the segment 0 \leq  x \leq  1 $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1a]</TD> <TD valign="top">  S.N. Bernshtein,  , ''Collected works'' , '''1''' , Moscow  (1952)  pp. 105–106</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top">  S.N. Bernshtein,  , ''Collected works'' , '''2''' , Moscow  (1954)  pp. 310–348</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.L. Goncharov,  "The theory of interpolation and approximation of functions" , Moscow  (1954)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.A. Baskakov,  "An instance of a sequence of linear positive operators in the space of continuous functions"  ''Dokl. Akad. Nauk SSSR'' , '''113''' :  2  (1957)  pp. 249–251  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  P.P. Korovkin,  "Linear operators and approximation theory" , Hindushtan Publ. Comp.  (1960)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  L.V. Kantorovich,  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''8'''  (1931)  pp. 1103–1115</TD></TR></table>
 
<table><TR><TD valign="top">[1a]</TD> <TD valign="top">  S.N. Bernshtein,  , ''Collected works'' , '''1''' , Moscow  (1952)  pp. 105–106</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top">  S.N. Bernshtein,  , ''Collected works'' , '''2''' , Moscow  (1954)  pp. 310–348</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.L. Goncharov,  "The theory of interpolation and approximation of functions" , Moscow  (1954)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.A. Baskakov,  "An instance of a sequence of linear positive operators in the space of continuous functions"  ''Dokl. Akad. Nauk SSSR'' , '''113''' :  2  (1957)  pp. 249–251  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  P.P. Korovkin,  "Linear operators and approximation theory" , Hindushtan Publ. Comp.  (1960)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  L.V. Kantorovich,  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''8'''  (1931)  pp. 1103–1115</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
There is also a multi-variable generalization: generalized Bernstein polynomials defined by the completely analogous formula
 
There is also a multi-variable generalization: generalized Bernstein polynomials defined by the completely analogous formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015730/b01573018.png" /></td> </tr></table>
+
$$
 +
B _ {\mathbf n }  (f, x _ {1} \dots x _ {k} ) =
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015730/b01573019.png" /></td> </tr></table>
+
$$
 +
= \
 +
\sum _ { i _ {1} = 0 } ^ { {n _ 1 } } \dots \sum
 +
_ {i _ {k} = 0 } ^ { {n _ k} } f \left (
 +
\frac{i _ {1} }{n _ {1} }
 +
\dots
 +
\frac{i _ {k} }{n _ {k} }
 +
\right ) \
 +
\left ( \begin{array}{c}
 +
n _ {1} \\
 +
i _ {1}
 +
\end{array}
 +
\right ) \dots
 +
\left ( \begin{array}{c}
 +
n _ {k} \\
 +
i _ {k}
 +
\end{array}
 +
\right ) \times
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015730/b01573020.png" /></td> </tr></table>
+
$$
 +
\times
 +
x _ {1} ^ {i _ {1} } (1 - x _ {1} ) ^ {n _ {1} - i _ {1} } \dots x _ {k} ^ {i _ {k} } (1 - x _ {k} ) ^ {n _ {k} - i _ {k} } .
 +
$$
  
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015730/b01573021.png" /> stands for the multi-index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015730/b01573022.png" />.
+
Here $  \mathbf n $
 +
stands for the multi-index $  \mathbf n = ( n _ {1} \dots n _ {k} ) $.
  
 
As in the one variable case these provide explicit polynomial approximants for the more-variable Weierstrass approximation and Stone–Weierstrass theorems. For the behaviour of Bernstein polynomials in the complex plane and applications to movement problems, cf. also [[#References|[a3]]].
 
As in the one variable case these provide explicit polynomial approximants for the more-variable Weierstrass approximation and Stone–Weierstrass theorems. For the behaviour of Bernstein polynomials in the complex plane and applications to movement problems, cf. also [[#References|[a3]]].

Latest revision as of 10:58, 29 May 2020


Algebraic polynomials defined by the formula

$$ B _ {n} (f; x ) = \ B _ {n} (x ) = $$

$$ = \ \sum _ { k=0 } ^ { n } f \left ( { \frac{k}{n} } \right ) \left ( \begin{array}{c} n \\ k \end{array} \right ) x ^ {k} (1-x) ^ {n-k} ,\ n = 1, 2 ,\dots . $$

Introduced by S.N. Bernshtein in 1912 (cf. ). The sequence of Bernstein polynomials converges uniformly to a function $ f $ on the segment $ 0 \leq x \leq 1 $ if $ f $ is continuous on this segment. For a function which is bounded by $ C $, $ 0 < C < 1 $, with a discontinuity of the first kind,

$$ B _ {n} (f; C) \rightarrow \ { \frac{f (C _ {-} ) + f (C _ {+} ) }{2} } . $$

The equation

$$ B _ {n} (f; c) - f (c) = \ \frac{f ^ { \prime\prime } (c)c(1-c) }{2n} + o \left ( \frac{1}{n} \right ) $$

is valid if $ f $ is twice differentiable at the point $ c $. If the $ k $- th derivative $ f ^ { (k) } $ of the function is continuous on the segment $ 0 \leq x \leq 1 $, the convergence

$$ B _ {n} ^ { (k) } (f; x) \rightarrow f ^ { (k) } (x) $$

is uniform on this segment. A study was made ([1b], [5]) of the convergence of Bernstein polynomials in the complex plane if $ f $ is analytic on the segment $ 0 \leq x \leq 1 $.

References

[1a] S.N. Bernshtein, , Collected works , 1 , Moscow (1952) pp. 105–106
[1b] S.N. Bernshtein, , Collected works , 2 , Moscow (1954) pp. 310–348
[2] V.L. Goncharov, "The theory of interpolation and approximation of functions" , Moscow (1954) (In Russian)
[3] V.A. Baskakov, "An instance of a sequence of linear positive operators in the space of continuous functions" Dokl. Akad. Nauk SSSR , 113 : 2 (1957) pp. 249–251 (In Russian)
[4] P.P. Korovkin, "Linear operators and approximation theory" , Hindushtan Publ. Comp. (1960) (Translated from Russian)
[5] L.V. Kantorovich, Izv. Akad. Nauk SSSR Ser. Mat. , 8 (1931) pp. 1103–1115

Comments

There is also a multi-variable generalization: generalized Bernstein polynomials defined by the completely analogous formula

$$ B _ {\mathbf n } (f, x _ {1} \dots x _ {k} ) = $$

$$ = \ \sum _ { i _ {1} = 0 } ^ { {n _ 1 } } \dots \sum _ {i _ {k} = 0 } ^ { {n _ k} } f \left ( \frac{i _ {1} }{n _ {1} } \dots \frac{i _ {k} }{n _ {k} } \right ) \ \left ( \begin{array}{c} n _ {1} \\ i _ {1} \end{array} \right ) \dots \left ( \begin{array}{c} n _ {k} \\ i _ {k} \end{array} \right ) \times $$

$$ \times x _ {1} ^ {i _ {1} } (1 - x _ {1} ) ^ {n _ {1} - i _ {1} } \dots x _ {k} ^ {i _ {k} } (1 - x _ {k} ) ^ {n _ {k} - i _ {k} } . $$

Here $ \mathbf n $ stands for the multi-index $ \mathbf n = ( n _ {1} \dots n _ {k} ) $.

As in the one variable case these provide explicit polynomial approximants for the more-variable Weierstrass approximation and Stone–Weierstrass theorems. For the behaviour of Bernstein polynomials in the complex plane and applications to movement problems, cf. also [a3].

References

[a1] P.J. Davis, "Interpolation and approximation" , Dover, reprint (1975) pp. 108–126
[a2] T.J. Rivlin, "An introduction to the approximation of functions" , Dover, reprint (1981)
[a3] G.G. Lorentz, "Bernstein polynomials" , Univ. Toronto Press (1953)
How to Cite This Entry:
Bernstein polynomials. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bernstein_polynomials&oldid=13289
This article was adapted from an original article by P.P. Korovkin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article