View source for Algebraic polynomial of best approximation

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A polynomial deviating least from a given function. More precisely, let a measurable function  $  f(x) $
be in  $  L _ {p} [a, b] $(
$  p \geq  1 $) 
and let  $  H _ {n} $
be the set of algebraic polynomials of degree not exceeding  $  n $. 
The quantity

$$ \tag{* }
E _ {n} (f) _ {p}  =  \inf _ {P _ {n} (x) \in
H _ {n} }  \| f (x) - P _ {n} (x) \| _ {L _ {p}  [ a , b ] }
$$

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  $  L _ {p} [a, b] $. 
Polynomials which deviate least from a given continuous function in the uniform metric ( $  p = \infty $) 
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  $  P _ {n}  ^ {0} (x) $
is an algebraic polynomial of best approximation in the uniform metric if and only if [[Chebyshev alternation|Chebyshev alternation]] occurs in the difference  $  f(x) - P _ {n}  ^ {0} (x) $; 
in this case such a polynomial is unique. If  $  p > 1 $, 
the algebraic polynomial of best approximation is unique due to the strict convexity of the space  $  L _ {p} $. 
If  $  p = 1 $, 
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  $  E _ {n} {(f) } _ {p} $
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  $  m $. 
If the number of variables  $  m \geq  2 $, 
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  $  E _ {n} (f) _ {p} $.

====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>