View source for Chebyshev alternation

A property of the difference between a continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021820/c0218201.png" /> on a closed set of real numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021820/c0218202.png" /> and a polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021820/c0218203.png" /> (in a [[Chebyshev system|Chebyshev system]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021820/c0218204.png" />) on an ordered sequence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021820/c0218205.png" /> points

<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/c/c021/c021820/c0218206.png" /></td> </tr></table>

such that

<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/c/c021/c021820/c0218207.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021820/c0218208.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021820/c0218209.png" />. The points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021820/c02182010.png" /> are called Chebyshev alternation points or points in Chebyshev alternation (cf. also [[Alternation, points of|Alternation, points of]]).



====Comments====
Points in Chebyshev alternation are also called Chebyshev points of alternation, and their set is also called an alternating set. See also (the references in) [[Alternation, points of|Alternation, points of]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.G. Lorentz,   "Approximation of functions" , Holt, Rinehart &amp; Winston  (1966)  pp. Chapt. 2, Sect. 6</TD></TR></table>