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
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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
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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=47770
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