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Quadratic HermitePadé approximation to the exponential function: a Riemann–Hilbert approach
 Constr. Approx
"... We investigate the asymptotic behavior of the polynomials p, q, r of degrees n in type I HermitePadé approximation to the exponential function, defined by p(z)e −z + q(z) + r(z)e z = O(z 3n+2) as z → 0. These polynomials are characterized by a Riemann– Hilbert problem for a 3 × 3 matrix valued func ..."
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Cited by 10 (4 self)
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We investigate the asymptotic behavior of the polynomials p, q, r of degrees n in type I HermitePadé approximation to the exponential function, defined by p(z)e −z + q(z) + r(z)e z = O(z 3n+2) as z → 0. These polynomials are characterized by a Riemann– Hilbert problem for a 3 × 3 matrix valued function. We use the DeiftZhou steepest descent method for Riemann–Hilbert problems to obtain strong uniform asymptotics for the scaled polynomials p(3nz), q(3nz), and r(3nz) in every domain in the complex plane. An important role is played by a threesheeted Riemann surface and certain measures and functions derived from it. Our work complements recent results of Herbert Stahl. Contents
Quadratic HermitePadé Polynomials Associated with the Exponential Function
 Electronic Trans. Num. Anal
, 2002
"... The asymptotic behavior of quadratic HermitePade polynomials pn ; qn ; rn 2 Pn associated with the exponential function is studied for n ! 1. These polynomials are de ned by pn (z) + qn (z)e + rn (z)e ) as z ! 0; (*) where O() denotes Landau's symbol. In the investigation analytic express ..."
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Cited by 5 (2 self)
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The asymptotic behavior of quadratic HermitePade polynomials pn ; qn ; rn 2 Pn associated with the exponential function is studied for n ! 1. These polynomials are de ned by pn (z) + qn (z)e + rn (z)e ) as z ! 0; (*) where O() denotes Landau's symbol. In the investigation analytic expressions are proved for the asymptotics of the polynomials, for the asymptotics of the remainder term in (*), and also for the arcs on which the zeros of the polynomials and of the remainder term cluster if the independent variable z is rescaled in an appropriate way. The asymptotic expressions are de ned with the help of an algebraic function of third degree and its associated Riemann surface. Among other possible applications, the results form the basis for the investigation of the convergence of quadratic HermitePad e approximants, which will be done in a followup paper.
Asymptotic Distributions of Zeros of Quadratic HermitePadé Polynomials Associated with the Exponential Function
"... The asymptotic distributions of zeros of the quadratic HermitePade polynomials pn ; qn ; rn 2 Pn associated with the exponential function are studied for n ! 1. The polynomials are de ned by the relation pn (z) + qn (z)e + rn (z)e 2z ) as z ! 0; (*) and they form the basis for quadratic H ..."
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Cited by 3 (1 self)
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The asymptotic distributions of zeros of the quadratic HermitePade polynomials pn ; qn ; rn 2 Pn associated with the exponential function are studied for n ! 1. The polynomials are de ned by the relation pn (z) + qn (z)e + rn (z)e 2z ) as z ! 0; (*) and they form the basis for quadratic HermitePade approximants to e . In order to achieve a dierentiated picture of the asymptotic behavior of the zeros, the independent variable z is rescaled in such a way that all zeros of the polynomials pn ; qn ; rn have nite cluster points as n ! 1. The asymptotic relations, which are proved, have a precision that is high enough to distinguish the positions of individual zeros. In addition to the zeros of the polynomials pn ; qn ; rn also the zeros of the remainder term of (*) are studied. The investigations complement asymptotic results obtained in [17].
Asymptotics of HermitePadé rational approximants for two analytic functions with separated pairs of branch points (case of genus 0)
, 2007
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MULTIPOINT TAYLOR EXPANSIONS OF ANALYTIC FUNCTIONS
, 2004
"... Abstract. Taylor expansions of analytic functions are considered with respect to several points, allowing confluence of any of them. Cauchytype formulas are given for coefficients and remainders in the expansions, and the regions of convergence are indicated. It is explained how these expansions ca ..."
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Cited by 1 (1 self)
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Abstract. Taylor expansions of analytic functions are considered with respect to several points, allowing confluence of any of them. Cauchytype formulas are given for coefficients and remainders in the expansions, and the regions of convergence are indicated. It is explained how these expansions can be used in deriving uniform asymptotic expansions of integrals. The method is also used for obtaining Laurent expansions in several points as well as TaylorLaurent expansions. 1.
Centrum voor Wiskunde en Informatica MAS Modelling, Analysis and Simulation Modelling, Analysis and Simulation
"... CWI's research has a themeoriented structure and is grouped into four clusters. Listed below are the names of the clusters and in parentheses their acronyms. ..."
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CWI's research has a themeoriented structure and is grouped into four clusters. Listed below are the names of the clusters and in parentheses their acronyms.