Chebyshev alternation

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

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

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