Some new properties of Chebyshev polynomials
The paper deals with the problem of the selection of optimal nodes for polynomial interpolation of data subject to bounded perturbations. Let \(\tau=\{\tau_i\}\) be a set of nodes \(\tau\subset [t_a, t_b],\;1\leq i\leq n\), \(t_a\leq \tau_1<\tau_2<...<\tau_n\leq t_b\), and let \(\Pi_n\) be the set of the \((n-1)\)th-degree polynomial that assumes alternatively the values \(+1\) and \(-1\) at the nodes \(\tau_i\). Denote by \(\tau_{opt}\) the family of all extrema of the \((n-1)\)-degree first kind Chebyshev polynomial \(T_{n-1}\), shifted to the interval \((t_a, t_b)\). The following theorems are proved and applyed to the problem of the selection of optimal nodes. Theorem 1: For any \(t\notin [t_a, t_b]\) the minimym \(| P(t)| \), \(P\in \Pi_n\), is obtained for \(\tau=\tau_{opt}\). Theorem 2: The set nodes \(\tau_{opt}\) minimized also all derivatives \(| P^{(i)} (t)| \), \(P\in \Pi_n\), \(1\leq i\leq n-1\) at any \(t\notin [t_a, t_b]\). In connection with Theorem 1 it should be pointed out that in 1918 the following statement of Bernstein was published: Theorem: Let \(\pi_{n, \tau}\) be the set of \((n-1)\)-degree polynomial \(P\) satisfies the conditions \(| P(\tau_i)| \leq 1,\;\;1\leq i\leq n.\) Then for \([t_a, t_b]=[-1,1]\) and any \(t>1\) the \(\min_{\tau\subset [-1,1]} \max_{P\in \pi_{n, \tau}} | P(t)| \) is obtained for \(\tau=\tau_{opt}\) [see, for example, Math. Ann. 79, 1--12 (1918; JFM 46.0417.01) or Collected Works of S. N. Bernstein, Vol. 1. Akad. Nauk USSR (1952; Zbl 0047.07301), pp. 253--264].
scientific article; zbMATH DE number 1472305
scientific article