Namespaces
Variants
Actions

Difference between revisions of "Markov criterion"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (tex encoded by computer)
m (fix tex)
 
Line 32: Line 32:
 
$$  
 
$$  
 
P _ {n}  ^ {*} ( x)  = \  
 
P _ {n}  ^ {*} ( x)  = \  
\sum _ { k= } 1 ^ { n }  
+
\sum _ { k= 1} ^ { n }  
 
c _ {k}  ^ {*} \phi _ {k} ( x)
 
c _ {k}  ^ {*} \phi _ {k} ( x)
 
$$
 
$$
Line 46: Line 46:
 
\int\limits _ { a } ^ { b }  \left |
 
\int\limits _ { a } ^ { b }  \left |
 
f ( x) -
 
f ( x) -
\sum _ { k= } 1 ^ { n }  c _ {k} \phi _ {k} ( x) \  
+
\sum _ { k= 1} ^ { n }  c _ {k} \phi _ {k} ( x) \  
 
\right |  d x =
 
\right |  d x =
 
$$
 
$$

Latest revision as of 19:58, 17 December 2020


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=47770
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