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A system of linearly independent functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021980/c0219801.png" /> in a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021980/c0219802.png" /> with the property that no non-trivial polynomial in this system has more than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021980/c0219803.png" /> distinct zeros. An example of a Chebyshev system in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021980/c0219804.png" /> is the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021980/c0219805.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021980/c0219806.png" />; its approximation properties in the uniform metric were first studied by P.L. Chebyshev [[#References|[1]]]. The term  "Chebyshev system"  was introduced by e theory of functions','../c/c025430.htm','Continuation method (to a parametrized family)','../c/c025520.htm','Euler–Lagrange equation','../e/e036510.htm','Fourier series','../f/f041090.htm','Functions of a real variable, theory of','../f/f042130.htm','Hilbert problems','../h/h120080.htm','Jackson inequality','../j/j054000.htm','Laplace theorem','../l/l057530.htm','Lebesgue constants','../l/l057800.htm','Limit theorems','../l/l058920.htm','Linear elliptic partial differential equation and system','../l/l059180.htm','Lyapunov theorem','../l/l061200.htm','Mathematical statistics','../m/m062710.htm','Minimal surface','../m/m063920.htm','Markov–Bernstein-type inequalities','../m/m110060.htm','Ornstein–Uhlenbeck process','../o/o070240.htm','Orthogonal polynomials','../o/o070340.htm','Plateau problem, multi-dimensional','../p/p072850.htm','Quasi-analytic class','../q/q076370.htm')" style="background-color:yellow;">S.N. Bernshtein [[#References|[2]]]. An arbitrary Chebyshev system inherits practically all approximation properties of the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021980/c0219807.png" />.
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A system of linearly independent functions $S=\{\phi_i\}_{i=1}^n$ in a space $C(Q)$ with the property that no non-trivial polynomial in this system has more than $n-1$ distinct zeros. An example of a Chebyshev system in $C[0,1]$ is the system $S_n^0=\{q^i\}_{i=0}^{n-1}$, $0\leq q\leq1$; its approximation properties in the uniform metric were first studied by P.L. Chebyshev [[#References|[1]]]. The term  "Chebyshev system"  was introduced by S.N. Bernshtein [[#References|[2]]]. An arbitrary Chebyshev system inherits practically all approximation properties of the system $S_n^0$.
  
The [[Chebyshev theorem|Chebyshev theorem]] and the [[De la Vallée-Poussin theorem|de la Vallée-Poussin theorem]] (on alternation) remain valid for Chebyshev systems; all methods developed for the approximate construction of algebraic polynomials of best uniform approximation apply equally well and the uniqueness theorem for polynomials of best uniform approximation is valid for Chebyshev systems (see also [[Haar condition|Haar condition]]; [[Chebyshev set|Chebyshev set]]). A compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021980/c0219808.png" /> admits a Chebyshev system of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021980/c0219809.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021980/c02198010.png" /> is homeomorphic to the circle or to a subset of it (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021980/c02198011.png" /> is not homeomorphic to the circle when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021980/c02198012.png" /> is even). In particular, there is no Chebyshev system on any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021980/c02198013.png" />-dimensional domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021980/c02198014.png" />, for example on a square [[#References|[3]]].
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The [[Chebyshev theorem|Chebyshev theorem]] and the [[De la Vallée-Poussin theorem|de la Vallée-Poussin theorem]] (on alternation) remain valid for Chebyshev systems; all methods developed for the approximate construction of algebraic polynomials of best uniform approximation apply equally well and the uniqueness theorem for polynomials of best uniform approximation is valid for Chebyshev systems (see also [[Haar condition|Haar condition]]; [[Chebyshev set|Chebyshev set]]). A compact set $Q$ admits a Chebyshev system of degree $n>1$ if and only if $Q$ is homeomorphic to the circle or to a subset of it ($Q$ is not homeomorphic to the circle when $n$ is even). In particular, there is no Chebyshev system on any $m$-dimensional domain $(m\geq2)$, for example on a square [[#References|[3]]].
  
As an example of a system that is not a Chebyshev system, one can mention the system consisting of splines (cf. [[Spline|Spline]]) of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021980/c02198015.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021980/c02198016.png" /> fixed knots <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021980/c02198017.png" />. In this case the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021980/c02198018.png" /> belongs to the system, and has infinitely many zeros. Lack of uniqueness makes the numerical construction of best approximations difficult.
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As an example of a system that is not a Chebyshev system, one can mention the system consisting of splines (cf. [[Spline|Spline]]) of degree $m$ with $n$ fixed knots $0<x_1<\ldots<x_n<1$. In this case the function $[\max(0,x-x_n)]^m$ belongs to the system, and has infinitely many zeros. Lack of uniqueness makes the numerical construction of best approximations difficult.
  
 
An important special case of a Chebyshev system is a [[Markov function system|Markov function system]].
 
An important special case of a Chebyshev system is a [[Markov function system|Markov function system]].

Latest revision as of 10:46, 15 August 2014

A system of linearly independent functions $S=\{\phi_i\}_{i=1}^n$ in a space $C(Q)$ with the property that no non-trivial polynomial in this system has more than $n-1$ distinct zeros. An example of a Chebyshev system in $C[0,1]$ is the system $S_n^0=\{q^i\}_{i=0}^{n-1}$, $0\leq q\leq1$; its approximation properties in the uniform metric were first studied by P.L. Chebyshev [1]. The term "Chebyshev system" was introduced by S.N. Bernshtein [2]. An arbitrary Chebyshev system inherits practically all approximation properties of the system $S_n^0$.

The Chebyshev theorem and the de la Vallée-Poussin theorem (on alternation) remain valid for Chebyshev systems; all methods developed for the approximate construction of algebraic polynomials of best uniform approximation apply equally well and the uniqueness theorem for polynomials of best uniform approximation is valid for Chebyshev systems (see also Haar condition; Chebyshev set). A compact set $Q$ admits a Chebyshev system of degree $n>1$ if and only if $Q$ is homeomorphic to the circle or to a subset of it ($Q$ is not homeomorphic to the circle when $n$ is even). In particular, there is no Chebyshev system on any $m$-dimensional domain $(m\geq2)$, for example on a square [3].

As an example of a system that is not a Chebyshev system, one can mention the system consisting of splines (cf. Spline) of degree $m$ with $n$ fixed knots $0<x_1<\ldots<x_n<1$. In this case the function $[\max(0,x-x_n)]^m$ belongs to the system, and has infinitely many zeros. Lack of uniqueness makes the numerical construction of best approximations difficult.

An important special case of a Chebyshev system is a Markov function system.

References

[1] P.L. Chebyshev, "Questions on smallest quantities connected with the approximate representation of functions (1859)" , Collected works , 2 , Moscow-Leningrad (1947) pp. 151–238 (In Russian)
[2] S.N. Bernshtein, "Extremal properties of polynomials and best approximation of continuous functions of a real variable" , 1 , Moscow-Leningrad (1937) (In Russian)
[3] J.C. Mairhuber, "On Haar's theorem concerning Chebyshev approximation problems having unique solutions" Proc. Amer. Math. Soc. , 7 : 4 (1956) pp. 609–615
[4] V.K. Dzyadyk, "Introduction to the theory of uniform approximation of functions by polynomials" , Moscow (1977) (In Russian)
[5] S. Karlin, V. Studden, "Tchebycheff systems with applications in analysis and statistics" , Interscience (1966)


Comments

References

[a1] G.G. Lorentz, "Approximation of functions" , Holt, Rinehart & Winston (1966) pp. Chapt. 2, Sect. 4
[a2] R. Zielke, "Discontinuous Čebyšev systems" , Springer (1979)
How to Cite This Entry:
Chebyshev system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Chebyshev_system&oldid=12169
This article was adapted from an original article by Yu.N. Subbotin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article