# Chebyshev alternation

From Encyclopedia of Mathematics

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