View source for Algebraic polynomial of best approximation

A polynomial deviating least from a given function. More precisely, let a measurable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011620/a0116201.png" /> be in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011620/a0116202.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011620/a0116203.png" />) and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011620/a0116204.png" /> be the set of algebraic polynomials of degree not exceeding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011620/a0116205.png" />. The quantity

<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/a/a011/a011620/a0116206.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

is called the [[Best approximation|best approximation]], while a polynomial for which the infimum is attained is known as an algebraic polynomial of best approximation in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011620/a0116207.png" />. Polynomials which deviate least from a given continuous function in the uniform metric (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011620/a0116208.png" />) were first encountered in the studies of P.L. Chebyshev (1852), who continued to study them in 1856 [[#References|[1]]]. The existence of algebraic polynomials of best approximation was established by E. Borel [[#References|[2]]]. Chebyshev proved that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011620/a0116209.png" /> is an algebraic polynomial of best approximation in the uniform metric if and only if [[Chebyshev alternation|Chebyshev alternation]] occurs in the difference <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011620/a01162010.png" />; in this case such a polynomial is unique. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011620/a01162011.png" />, the algebraic polynomial of best approximation is unique due to the strict convexity of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011620/a01162012.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011620/a01162013.png" />, it is not unique, but it has been shown by D. Jackson [[#References|[3]]] to be unique for continuous functions. The rate of convergence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011620/a01162014.png" /> to zero is given by Jackson's theorems (cf. [[Jackson theorem|Jackson theorem]]).

In a manner similar to (*) an algebraic polynomial of best approximation is defined for functions in a large number of unknowns, say <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011620/a01162015.png" />. If the number of variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011620/a01162016.png" />, an algebraic polynomial of best approximation in the uniform metric is, in general, not unique.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.L. Chebyshev,   "Questions on smallest quantities connected with the approximate representation of functions (1859)" , ''Collected works'' , '''2''' , Moscow-Leningrad  (1947)  pp. 478; 152–236  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E. Borel,   "Leçons sur les fonctions de variables réelles et les développements en séries de polynômes" , Gauthier-Villars  (1905)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  D. Jackson,   "A general class of problems in approximation"  ''Amer. J. Math.'' , '''46'''  (1924)  pp. 215–234</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.L. Garkavi,   "The theory of approximation in normed linear spaces"  ''Itogi Nauk. Mat. Anal. 1967''  (1969)  pp. 75–132  (In Russian)</TD></TR></table>



====Comments====
Instead of the long phrase  "algebraic polynomial of best approximation"  one also uses the shorter phrase  "best algebraic approximationbest algebraic approximation" , which is not to be confused with the phrase  "best approximation"  for the least error <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011620/a01162017.png" />.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N.I. [N.I. Akhiezer] Achiezer,   "Theory of approximation" , F. Ungar  (1956)  (Translated from Russian)</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><TR><TD valign="top">[a3]</TD> <TD valign="top">  G.W. Meinardus,   "Approximation von Funktionen und ihre numerische Behandlung" , Springer  (1964)  pp. Chapt. 1, §5</TD></TR></table>