Markov function system

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A system $\{\phi_\nu(x)\}_{\nu=1}^n$ ($n\leq\infty$) of linearly independent real-valued continuous functions defined on a finite interval $[a,b]$ and satisfying the condition: For any finite $k\leq n$ the functions $\phi_1(x),\ldots,\phi_k(x)$ form a Chebyshev system on $(a,b)$.

Examples of Markov function systems are:

a) $1,x,x^2,\ldots,$ on any interval $[a,b]$;

b) $1,\cos x,\cos2x,\ldots,$ on $[0,\pi]$;

c) $\sin x,\sin2x,\ldots,$ on $[0,\pi]$.


[1] N.I. [N.I. Akhiezer] Achiezer, "Theory of approximation" , F. Ungar (1956) (Translated from Russian)



[a1] E.W. Cheney, "Introduction to approximation theory" , Chelsea, reprint (1982)
[a2] W.J. Studden, "Tchebycheff systems: with applications in analysis and statistics" , Wiley (1966)
[a3] H.S. Shapiro, "Topics in approximation theory" , Springer (1971)
[a4] I.M. Singer, "Best approximation in normed linear spaces by elements of linear subspaces" , Springer (1970)
How to Cite This Entry:
Markov function system. 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