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A lower bound for the norm of the minimal residual polynomial. (English) Zbl 1239.41005
The paper deals with some interesting properties of the minimal residual polynomial of degree at most \(n\) on \(S\), where \(S\) is a compact infinite set in the complex plane which does not contain the origin. First, some essential properties of the minimal residual polynomial on a real set are provided (Corollary 1, Lemma 2, Corollary 2). The main result of the paper is contained in Theorem 2, where the author gives a refinement for the inequality verified by the norm \(L_n(S)\) of the minimal residual polynomial in the case that \(S\) is a union of a finite number of real intervals. As a consequence, he obtains a slight refinement of the Bernstein-Walsh lemma (Corollary 3).

MSC:
41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
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