Direct and Inverse Results on Row Sequences of Simultaneous Padé–Faber Approximants

  • Nattapong Bosuwan
  • Guillermo López LagomasinoEmail author


Given a vector function \(\mathbf F =(F_1,\ldots ,F_d),\) analytic on a neighborhood of some compact subset E of the complex plane with simply connected complement, we define a sequence of vector rational functions with common denominator in terms of the expansions of the components \(F_k, k=1,\ldots ,d,\) with respect to the sequence of Faber polynomials associated with E. Such sequences of vector rational functions are analogous to row sequences of type II Hermite–Padé approximation. We give necessary and sufficient conditions for the convergence with geometric rate of the common denominators of the sequence of vector rational functions so constructed. The exact rate of convergence of these denominators is provided and the rate of convergence of the approximants is estimated. It is shown that the common denominators of the approximants detect the poles of the system of functions “closest” to E and their order.


Montessus de Ballore’s theorem Faber polynomials simultaneous approximation Hermite–Padé approximation rate of convergence inverse results 

Mathematics Subject Classification

Primary 30E10 41A21 41A28 Secondary 41A25 41A27 



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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceMahidol UniversityBangkokThailand
  2. 2.Centre of Excellence in Mathematics, CHEBangkokThailand
  3. 3.Mathematics DepartmentUniversidad Carlos III de MadridLeganésSpain

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