Bernstein interpolation method

A sequence of algebraic polynomials converging uniformly on to a function that is continuous on this interval. More precisely, Bernstein's interpolation method is a sequence of algebraic polynomials

where the

are the Chebyshev polynomials; the

are the interpolation nodes; and

if is an arbitrary positive integer, , , , otherwise

The ratio between the degree of the polynomial and the number of points at which equals is , which tends to as ; if 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 , , for bounded functions on . This method is given by the Bernstein polynomials. The sequence of Bernstein polynomials constructed for a bounded function on converges to at each point of continuity of . If is continuous on , the sequence converges uniformly (to ) on . If is differentiable, (at each point of continuity of ), 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