# Markov criterion

From Encyclopedia of Mathematics

*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 . It was established by A.A. Markov in 1898 (see [1]). Let , , be a system of linearly independent functions continuous on the interval , and let the continuous function change sign at the points in and be such that

If the polynomial

has the property that the difference changes sign at the points , and only at those points, then is the polynomial of best integral approximation to and

For the system on , can be taken to be ; for the system , , can be taken to be ; and for the system , , one can take .

#### 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