Difference between revisions of "Markov function system"
From Encyclopedia of Mathematics
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| − | A system | + | {{TEX|done}} |
| + | 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|Chebyshev system]] on $(a,b)$. | ||
Examples of Markov function systems are: | Examples of Markov function systems are: | ||
| − | a) | + | a) $1,x,x^2,\ldots,$ on any interval $[a,b]$; |
| − | b) | + | b) $1,\cos x,\cos2x,\ldots,$ on $[0,\pi]$; |
| − | c) | + | c) $\sin x,\sin2x,\ldots,$ on $[0,\pi]$. |
====References==== | ====References==== | ||
Latest revision as of 06:03, 5 August 2014
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 $[0,\pi]$.
References
| [1] | N.I. [N.I. Akhiezer] Achiezer, "Theory of approximation" , F. Ungar (1956) (Translated from Russian) |
Comments
References
| [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: http://encyclopediaofmath.org/index.php?title=Markov_function_system&oldid=12310
Markov function system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Markov_function_system&oldid=12310
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