Nevanlinna, Olavi Multicentric holomorphic calculus. (English) Zbl 1337.47021 Comput. Methods Funct. Theory 12, No. 1, 45-65 (2012). Let \(A\) be a complex unital Banach algebra. For \(a\in A\), let \(\widehat{\sigma(a)}\) denote the polynomially convex hull of the spectrum of \(a\). Let \(\Omega\subset \mathbb{C}\) be an open set such that \(\widehat{\sigma(a)}\subset\Omega\). If \(f\) is holomorphic in \(\Omega\) and \(\Gamma\subset\Omega\) is a contour surrounding \(\widehat{\sigma(a)}\), then \[ f(a) = \frac{1}{2\pi i} \int_{\Gamma} f(\lambda) (\lambda-a)^{-1} d\lambda. \]The author constructs a practical power series calculus based on approximations of the resolvent \((\lambda-a)^{-1}\) by rational functions. This approach was outlined by the author in [Numer. Funct. Anal. Optim. 30, No. 9–10, 1025–1047 (2009; Zbl 1188.47005)]. Reviewer: Evgueni Doubtsov (St. Petersburg) (MR2977278) Cited in 4 Documents MSC: 47A60 Functional calculus for linear operators 46H30 Functional calculus in topological algebras 47A10 Spectrum, resolvent 30B99 Series expansions of functions of one complex variable 32E30 Holomorphic, polynomial and rational approximation, and interpolation in several complex variables; Runge pairs 34L16 Numerical approximation of eigenvalues and of other parts of the spectrum of ordinary differential operators Keywords:Jacobi series; spectrum; holomorphic functional calculus; polynomial numerical hull Citations:Zbl 1188.47005 PDFBibTeX XMLCite \textit{O. Nevanlinna}, Comput. Methods Funct. Theory 12, No. 1, 45--65 (2012; Zbl 1337.47021) Full Text: DOI References: [1] J.-P. Berrut and L. N. Trefethen, Barycentric Lagrange interpolation, SIAM Review 46 no.3 (2004), 501–517. · Zbl 1061.65006 · doi:10.1137/S0036144502417715 [2] J. H. Curtiss, On the Jacobi series, Trans. Amer. Math. Soc. 49 no.3 (1941), 467–501. · Zbl 0024.41803 · doi:10.1090/S0002-9947-1941-0004299-2 [3] M. Fekete, Uber die Verallgemeinerung der Picard-Landauschen und Picard-Schottkyschen Sätze auf Reihen, die nach Potenzen eines Polynoms fortschreiten und Polynome niedrigeren Grades zu Koeffizienten haben, Math. Ann. 106 (1932), 595–616. · Zbl 0004.35703 · doi:10.1007/BF01455903 [4] B. R. Gelbaum, Modern Real and Complex Analysis, John Wiley Sons, Inc., 1995. [5] C. G. J. Jacobi, Über Reihenentwicklungen, welche nach den Potenzen eines gegebenen Polynoms fortschreiten, und zu Coeffizienten Polynome eines niedereren Grades haben, J. Reine Angew. Math. 53 (1856), 103–126. · ERAM 053.1383cj [6] A. Kienast, Über die Darstellung der analytischen Funktionen durch Reihen, die nach Potenzen eines Polynomes fortschreiten und Polynome eines niedereren Grades zu Koeffizienten haben, Inaugural Dissertation, Zürich, 1906. · JFM 37.0447.04 [7] A. I. Markushevich, Theory of Functions of a Complex Variable, Chelsea Publ. Co, New York, Second ed. 1977, Vol 2, pp. 75–79. [8] O. Nevanlinna, Convergence of Iterations for Linear Equations, Birkhäuser, Basel, 1993. · Zbl 0846.47008 [9] O. Nevanlinna, Hessenberg matrices in Krylov subspaces and the computation of the spectrum, Numer. Funct. Anal. Optim. 16 no.3–4 (1995), 443–473. · Zbl 0837.65056 · doi:10.1080/01630569508816627 [10] –, Meromorphic Functions and Linear Algebra, AMS Fields Institute Monograph 18, 2003. · Zbl 1156.30301 [11] O. Nevanlinna, Computing the spectrum and representing the resolvent, Numer. Funct. Anal. Optim. 30 no.9–10 (2009), 1025–1047. · Zbl 1188.47005 · doi:10.1080/01630560903393162 [12] Th. Ransford, Potential Theory in the Complex Plane, London Math. Soc. Student Texts 28, Cambridge Univ. Press, 1995. · Zbl 0828.31001 [13] B. Schweizer, On the Jacobi Series, Aequationes Math. 34 (1987), 186–194. · Zbl 0629.41018 · doi:10.1007/BF01830670 [14] J. L. Walsh, Interpolation and Approximation by Rational Functions in the Complex Domain, AMS Colloquium Publ. Vol XX, Fifth ed. 1969, pp. 54–56. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.