Markov criterion

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)

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)