Chebyshev system

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)

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)