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Chebyshev alternation

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A property of the difference between a continuous function $f(x)$ on a closed set of real numbers $Q$ and a polynomial $P_n(x)$ (in a Chebyshev system $\{\phi_k(x)\}_0^n$) on an ordered sequence of $n+2$ points

$$\{x_i\}_0^{n+1}\subset Q,\quad x_0<\dotsb<x_{n+1},$$

such that

$$f(x_i)-P_n(x_i)=(-1)^i\epsilon\|f(x)-P_n(x)\|_{C(Q)},$$

where $\epsilon=1$ or $-1$. The points $\{x_i\}_0^{n+1}$ are called Chebyshev alternation points or points in Chebyshev alternation (cf. also Alternation, points of).


Comments

Points in Chebyshev alternation are also called Chebyshev points of alternation, and their set is also called an alternating set. See also (the references in) Alternation, points of.

References

[a1] G.G. Lorentz, "Approximation of functions" , Holt, Rinehart & Winston (1966) pp. Chapt. 2, Sect. 6
How to Cite This Entry:
Chebyshev alternation. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Chebyshev_alternation&oldid=44592
This article was adapted from an original article by Yu.N. Subbotin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article