Bernstein polynomials

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)