View source for Bernstein interpolation method

A sequence of algebraic polynomials converging uniformly on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015710/b0157101.png" /> to a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015710/b0157102.png" /> that is continuous on this interval. More precisely, Bernstein's interpolation method is a sequence of algebraic polynomials

<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/b015710/b0157103.png" /></td> </tr></table>

where the

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are the [[Chebyshev polynomials|Chebyshev polynomials]]; the

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are the interpolation nodes; and

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if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015710/b0157107.png" /> is an arbitrary positive integer, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015710/b0157108.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015710/b0157109.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015710/b01571010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015710/b01571011.png" /> otherwise

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The ratio between the degree of the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015710/b01571013.png" /> and the number of points at which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015710/b01571014.png" /> equals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015710/b01571015.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015710/b01571016.png" />, which tends to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015710/b01571017.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015710/b01571018.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015710/b01571019.png" /> 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====
<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====
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015710/b01571020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015710/b01571021.png" />, for bounded functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015710/b01571022.png" />. This method is given by the [[Bernstein polynomials|Bernstein polynomials]]. The sequence of Bernstein polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015710/b01571023.png" /> constructed for a bounded function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015710/b01571024.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015710/b01571025.png" /> converges to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015710/b01571026.png" /> at each point of continuity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015710/b01571027.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015710/b01571028.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015710/b01571029.png" /> is continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015710/b01571030.png" />, the sequence converges uniformly (to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015710/b01571031.png" />) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015710/b01571032.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015710/b01571033.png" /> is differentiable, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015710/b01571034.png" /> (at each point of continuity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015710/b01571035.png" />), 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]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.J. Davis,   "Interpolation and approximation" , Dover, reprint  (1975)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E.W. Cheney,   "Introduction to approximation theory" , Chelsea, reprint  (1982)  pp. 203ff</TD></TR></table>