Bernstein polynomials

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Algebraic polynomials defined by the formula

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

The equation

is valid if is twice differentiable at the point . If the -th derivative of the function is continuous on the segment , the convergence

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


[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


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

Here stands for the multi-index .

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].


[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. P.P. Korovkin (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098