Markov criterion

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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 .


[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)



[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:
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