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''for best integral approximation''

A theorem which in some cases enables one to give effectively the polynomial and the error of best integral approximation of a function  $  f $. 
It was established by A.A. Markov in 1898 (see [[#References|[1]]]). Let  $  \{ \phi _ {k} ( x) \} $, 
$  k = 1 \dots n $, 
be a system of linearly independent functions continuous on the interval  $  [ a , b ] $, 
and let the continuous function  $  \psi $
change sign at the points  $  x _ {1} < \dots < x _ {r} $
in  $  ( a , b ) $
and be such that

$$ 
\int\limits _ { a } ^ { b } 
\phi _ {k} ( x)   \mathop{\rm sgn}  \psi ( x)  d x  =  0 ,\ \ 
k = 1 \dots n .
$$

If the polynomial

$$ 
P _ {n}  ^ {*} ( x)  = \ 
\sum _ { k= } 1 ^ { n } 
c _ {k}  ^ {*} \phi _ {k} ( x)
$$

has the property that the difference  $  f - P _ {n}  ^ {*} $
changes sign at the points  $  x _ {1} \dots x _ {r} $, 
and only at those points, then  $  P _ {n}  ^ {*} $
is the polynomial of best integral approximation to  $  f $
and

$$ 
\inf _ {\{ c _ {k} \} } \ 
\int\limits _ { a } ^ { b }  \left |
f ( x) -
\sum _ { k= } 1 ^ { n }  c _ {k} \phi _ {k} ( x) \ 
\right |  d x =
$$

$$ 
= \ 
\int\limits _ { a } ^ { b }  \left | f ( x) - P _ {n}  ^ {*} ( x) \right |  d x
 =  \left | \int\limits _ { a } ^ { b }  f ( x)   \mathop{\rm sgn}  \psi ( x)  d x \right | .
$$

For the system  $  \{ 1 , \cos  x \dots \cos  n x \} $
on  $  [ 0 , \pi ] $, 
$  \psi $
can be taken to be  $  \cos  ( n + 1) x $; 
for the system  $  \{ \sin  x \dots \sin  n x \} $, 
$  0 \leq  x \leq  \pi $, 
$  \psi $
can be taken to be  $  \sin  ( n + 1 ) x $; 
and for the system  $  \{ 1 , x \dots x  ^ {n} \} $, 
$  - 1 \leq  x \leq  1 $, 
one can take  $  \psi ( x) = \sin ( ( n + 2 )   \mathop{\rm arc}  \cos  x ) $.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.A. Markov,   "Selected works" , Moscow-Leningrad  (1948)  (In Russian)</TD></TR><TR><TD valign="top">[2]</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">[3]</TD> <TD valign="top">  I.K. Daugavet,   "Introduction to the theory of approximation of functions" , Leningrad  (1977)  (In Russian)</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.W. Cheney,   "Introduction to approximation theory" , Chelsea, reprint  (1982)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M.W. Müller,   "Approximationstheorie" , Akad. Verlagsgesellschaft  (1978)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J.R. Rice,   "The approximation of functions" , '''1. Linear theory''' , Addison-Wesley  (1964)</TD></TR></table>