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''for best integral approximation''
 
''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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062440/m0624401.png" />. It was established by A.A. Markov in 1898 (see [[#References|[1]]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062440/m0624402.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062440/m0624403.png" />, be a system of linearly independent functions continuous on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062440/m0624404.png" />, and let the continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062440/m0624405.png" /> change sign at the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062440/m0624406.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062440/m0624407.png" /> and be such that
+
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
  
<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/m/m062/m062440/m0624408.png" /></td> </tr></table>
+
$$
 +
\int\limits _ { a } ^ { b }
 +
\phi _ {k} ( x)  \mathop{\rm sgn}  \psi ( x)  d x  = 0 ,\ \
 +
k = 1, \dots, n .
 +
$$
  
 
If the polynomial
 
If the polynomial
  
<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/m/m062/m062440/m0624409.png" /></td> </tr></table>
+
$$
 +
P _ {n}  ^ {*} ( x)  = \
 +
\sum _ { k= 1}  ^ { n }
 +
c _ {k}  ^ {*} \phi _ {k} ( x)
 +
$$
  
has the property that the difference <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062440/m06244010.png" /> changes sign at the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062440/m06244011.png" />, and only at those points, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062440/m06244012.png" /> is the polynomial of best integral approximation to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062440/m06244013.png" /> and
+
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
  
<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/m/m062/m062440/m06244014.png" /></td> </tr></table>
+
$$
 +
\inf _ {\{ c _ {k} \} } \
 +
\int\limits _ { a } ^ { b }  \left |
 +
f ( x) -
 +
\sum _ { k= 1} ^ { n }  c _ {k} \phi _ {k} ( x) \
 +
\right |  d x =
 +
$$
  
<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/m/m062/m062440/m06244015.png" /></td> </tr></table>
+
$$
 +
= \
 +
\int\limits _ { a } ^ { b }  \left | f ( x) - P _ {n}  ^ {*} ( x) \right |  d x
 +
= \left | \int\limits _ { a } ^ { b }  f ( x)  \mathop{\rm sgn}  \psi ( x)  d x \right | .
 +
$$
  
For the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062440/m06244016.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062440/m06244017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062440/m06244018.png" /> can be taken to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062440/m06244019.png" />; for the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062440/m06244020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062440/m06244021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062440/m06244022.png" /> can be taken to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062440/m06244023.png" />; and for the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062440/m06244024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062440/m06244025.png" />, one can take <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062440/m06244026.png" />.
+
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====
 
====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>
 
<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====
 
====Comments====
 
  
 
====References====
 
====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>
 
<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>

Latest revision as of 12:21, 18 February 2022


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 [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

[1] A.A. Markov, "Selected works" , Moscow-Leningrad (1948) (In Russian)
[2] N.I. [N.I. Akhiezer] Achiezer, "Theory of approximation" , F. Ungar (1956) (Translated from Russian)
[3] I.K. Daugavet, "Introduction to the theory of approximation of functions" , Leningrad (1977) (In Russian)

Comments

References

[a1] E.W. Cheney, "Introduction to approximation theory" , Chelsea, reprint (1982)
[a2] M.W. Müller, "Approximationstheorie" , Akad. Verlagsgesellschaft (1978)
[a3] J.R. Rice, "The approximation of functions" , 1. Linear theory , Addison-Wesley (1964)
How to Cite This Entry:
Markov criterion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Markov_criterion&oldid=17793
This article was adapted from an original article by N.P. KorneichukV.P. Motornyi (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article