Alternation, points of

A sequence of points

$$\{x_i\}_0^{n+1}\subset Q,\quad a\leq x_0<x_1<\ldots<x_{n+1}\leq b,$$

at which the difference $\alpha_i=f(x_i)-P_n(x_i)$ assumes non-zero values of alternating signs. Here $f(x)$ is a continuous function on the closed set $Q\subset[a,b]$ and $P_n(x)$ is an algebraic polynomial of degree not exceeding $n$. In a similar manner the concept of points of alternation is introduced for polynomials in a Chebyshev system of functions $\{s_k(x)\}_0^n$ (which satisfy the Haar condition). If, in this situation, all absolute values of $\alpha_i$ are equal to

$$\max_{x\in Q}|f(x)-P_n(x)|,$$

then the points $\{x_i\}_0^{n+1}$ are called Chebyshev points of alternation. Points of alternation play an important role in the theory of approximation of functions. E.g., the de la Vallée-Poussin theorem (the alternation theorem) and the Chebyshev criterion (cf. Chebyshev alternation) are formulated in terms of points of alternation. Points of alternation are also employed in constructing polynomials of best approximation.

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A sequence of Chebyshev points of alternation is also called an alternating set [a1], Chapt. 1.

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