View source for 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|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|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|Alternation, points of]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.G. Lorentz,   "Approximation of functions" , Holt, Rinehart &amp; Winston  (1966)  pp. Chapt. 2, Sect. 6</TD></TR></table>