Difference between revisions of "Markov criterion"
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A theorem which in some cases enables one to give effectively the polynomial and the error of best integral approximation of a function $ f $. | 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) \} $, | It was established by A.A. Markov in 1898 (see [[#References|[1]]]). Let $ \{ \phi _ {k} ( x) \} $, | ||
| − | $ k = 1 \dots n $, | + | $ k = 1, \dots, n $, |
be a system of linearly independent functions continuous on the interval $ [ a , b ] $, | be a system of linearly independent functions continuous on the interval $ [ a , b ] $, | ||
and let the continuous function $ \psi $ | and let the continuous function $ \psi $ | ||
| Line 25: | Line 25: | ||
\int\limits _ { a } ^ { b } | \int\limits _ { a } ^ { b } | ||
\phi _ {k} ( x) \mathop{\rm sgn} \psi ( x) d x = 0 ,\ \ | \phi _ {k} ( x) \mathop{\rm sgn} \psi ( x) d x = 0 ,\ \ | ||
| − | k = 1 \dots n . | + | k = 1, \dots, n . |
$$ | $$ | ||
| Line 32: | Line 32: | ||
$$ | $$ | ||
P _ {n} ^ {*} ( x) = \ | P _ {n} ^ {*} ( x) = \ | ||
| − | \sum _ { k= } | + | \sum _ { k= 1} ^ { n } |
c _ {k} ^ {*} \phi _ {k} ( x) | c _ {k} ^ {*} \phi _ {k} ( x) | ||
$$ | $$ | ||
has the property that the difference $ f - P _ {n} ^ {*} $ | has the property that the difference $ f - P _ {n} ^ {*} $ | ||
| − | changes sign at the points $ x _ {1} \dots x _ {r} $, | + | changes sign at the points $ x _ {1}, \dots, x _ {r} $, |
and only at those points, then $ P _ {n} ^ {*} $ | and only at those points, then $ P _ {n} ^ {*} $ | ||
is the polynomial of best integral approximation to $ f $ | is the polynomial of best integral approximation to $ f $ | ||
| Line 46: | Line 46: | ||
\int\limits _ { a } ^ { b } \left | | \int\limits _ { a } ^ { b } \left | | ||
f ( x) - | f ( x) - | ||
| − | \sum _ { k= } | + | \sum _ { k= 1} ^ { n } c _ {k} \phi _ {k} ( x) \ |
\right | d x = | \right | d x = | ||
$$ | $$ | ||
| Line 56: | Line 56: | ||
$$ | $$ | ||
| − | For the system $ \{ 1 , \cos x \dots \cos n x \} $ | + | For the system $ \{ 1 , \cos x, \dots, \cos n x \} $ |
on $ [ 0 , \pi ] $, | on $ [ 0 , \pi ] $, | ||
$ \psi $ | $ \psi $ | ||
can be taken to be $ \cos ( n + 1) x $; | can be taken to be $ \cos ( n + 1) x $; | ||
| − | for the system $ \{ \sin x \dots \sin n x \} $, | + | for the system $ \{ \sin x, \dots, \sin n x \} $, |
$ 0 \leq x \leq \pi $, | $ 0 \leq x \leq \pi $, | ||
$ \psi $ | $ \psi $ | ||
can be taken to be $ \sin ( n + 1 ) x $; | can be taken to be $ \sin ( n + 1 ) x $; | ||
| − | and for the system $ \{ 1 , x \dots x ^ {n} \} $, | + | and for the system $ \{ 1 , x, \dots, x ^ {n} \} $, |
$ - 1 \leq x \leq 1 $, | $ - 1 \leq x \leq 1 $, | ||
one can take $ \psi ( x) = \sin ( ( n + 2 ) \mathop{\rm arc} \cos x ) $. | one can take $ \psi ( x) = \sin ( ( n + 2 ) \mathop{\rm arc} \cos x ) $. | ||
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) |
Markov criterion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Markov_criterion&oldid=47770