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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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##### References:
 [1] Achieser, N.I.: Über einige Funktionen, welche in zwei gegebenen Intervallen am wenigsten von Null abweichen. I. Bull. Acad. Sci. URSS 7(9), 1163–1202 (1932) · JFM 58.1065.02 [2] Aptekarev, A.I.: Asymptotic properties of polynomials orthogonal on a system of contours, and periodic motions of Toda chains. Math. USSR-Sb. 53, 233–260 (1986) · Zbl 0608.42016 [3] Driscoll, T.A., Toh, K.-C., Trefethen, L.N.: From potential theory to matrix iterations in six steps. SIAM Rev. 40, 547–578 (1998) (electronic) · Zbl 0930.65020 [4] Eiermann, M., Li, X., Varga, R.S.: On hybrid semi-iterative methods. SIAM J. Numer. Anal. 26, 152–168 (1989) · Zbl 0669.65020 [5] Fischer, B.: Chebyshev polynomials for disjoint compact sets. Constr. Approx. 8, 309–329 (1992) · Zbl 0778.41012 [6] Fischer, B.: Polynomial Based Iteration Methods for Symmetric Linear Systems. Wiley, New York (1996) · Zbl 0852.65031 [7] Kuijlaars, A.B.J.: Convergence analysis of Krylov subspace iterations with methods from potential theory. SIAM Rev. 48, 3–40 (2006) (electronic) · Zbl 1092.65031 [8] Peherstorfer, F.: Minimal polynomials for compact sets of the complex plane. Constr. Approx. 12, 481–488 (1996) · Zbl 0878.41003 [9] Peherstorfer, F.: Inverse images of polynomial mappings and polynomials orthogonal on them. J. Comput. Appl. Math. 153, 371–385 (2003) · Zbl 1034.30002 [10] Ransford, T.: Potential Theory in the Complex Plane. London Mathematical Society Student Texts, vol. 28. Cambridge University Press, Cambridge (1995) · Zbl 0828.31001 [11] Schiefermayr, K.: Estimates for the asymptotic convergence factor of two intervals. J. Comput. Appl. Math. (2010) · Zbl 1231.41012 [12] Schiefermayr, K.: A lower bound for the minimum deviation of the Chebyshev polynomial on a compact real set. East J. Approx. 14, 223–233 (2008) · Zbl 1217.41031 [13] Totik, V.: The norm of minimal polynomials on several intervals. J. Approx. Theory (2010) · Zbl 1221.41018 [14] Totik, V.: Polynomial inverse images and polynomial inequalities. Acta Math. 187, 139–160 (2001) · Zbl 0997.41005 [15] Totik, V.: How to prove results for polynomials on several intervals? In: Approximation Theory, DARBA, Sofia, 2002, pp. 397–410 · Zbl 1034.41014 [16] Totik, V.: Chebyshev constants and the inheritance problem. J. Approx. Theory 160, 187–201 (2009) · Zbl 1190.41002 [17] Widom, H.: Extremal polynomials associated with a system of curves in the complex plane. Adv. Math. 3, 127–232 (1969) · Zbl 0183.07503
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