On uniform convergence of diagonal multipoint Padé approximants for entire functions
type: Item
instance of: scholarly article
title: On uniform convergence of diagonal multipoint Padé approximants for entire functions
zbMATH Open document ID: 1407.41008
published in: Constructive Approximation
publication date: 2019-01-10T00:00:00Z
review text: In this very nice paper the author gives important new results on convergence of diagonal sequences of multipoint Padé approximants, connected with the \textit{Baker-Gammel-Wills conjecture}: Let \(f\) be meromorphic in \({\mathcal B}_1\) and analytic at \(z=0\). Then there is a subsequence \(\{[n/n]\}_{n\in{\mathcal S}}\) of \(\{[n/n]\}_{n\geq 1}\) that converges uniformly to \(f\) in compact subsets of \({\mathcal B}_1\) omitting poles of \(f\) The new results are: Theorem 1.1 There is an \(F_{\sigma}\) subset \(\mathcal E\) of the first category of \(\mathcal A\) (\(\mathcal A\) the set of entire functions with metric defined using the coefficients of the power series expansions), such that for \(f\in{\mathcal A}\setminus{\mathcal E}\), there is an infinite subsequence \(\mathcal S\) of positive integers with the following property: given any \(r>0\) and for any \(n\in{\mathcal S}\), multipoint Padé approximants \(R_n\) to \(f\) of type \((n,n)\) formed from interpolation points \(\Lambda\subset {\mathcal B}_r\), we have \[ \lim_{n\rightarrow\infty, n\in{\mathcal S}}\,R_n(z)=f(z),\tag{1} \] uniformly in compact subsets of the plane. Theorem 1.2 Assume that \[ f(z)=\sum_{j=0}^{\infty}\,a_jz^j ,\tag{2} \] where \(a_j\not= 0\) for \(j\geq 0\) and for some fixed \(J\), and for \(j\geq J\), \[ \left|\frac{a_{j-1}a_{j+1}}{a_j^2}\right|\leq \chi^2, \] where \(\chi<\rho_0\) and \(\rho_0=0.4599\cdots\) is the positive root of the equation \[ \sum_{j=1}^{\infty}\,\rho^{j^2}=\frac{1}{2}. \] Let \(r>0\). For \(n\geq 1\), let \(R_n\) denote a multipoint Padé approximant to \(f\) formed with interpolation points in \({\mathcal B}_r\). Then uniformly in compact subsets of the plane \[ \lim_{n\rightarrow\infty}\,R_n(z)=f(z). \] Theorem 1.3 Assume that \(f\) is given by \((2)\), where \(a_j\not= 0\) for \(j\geq 0\) and for some \(|q|<1\), \[ \lim_{j\rightarrow\infty}\,\frac{a_{j-1}a_{j+1}}{a_j^2}=q. \] Then the conclusion of Theorem 1.2 remains valid. Theorem 1.4 Assume for \(f\)…
reviewed by: Marcel G. de Bruin
Mathematics Subject Classification ID: 41A21
Mathematics Subject Classification ID: 41A20
Mathematics Subject Classification ID: 30E10
zbMATH DE Number: 6999349
zbMATH Keywords: Padé approximation
zbMATH Keywords: multipoint Padé approximants
zbMATH Keywords: spurious poles
zbMATH Keywords: Baker-Gammel-Wills conjecture
author: Doron S. Lubinsky
MaRDI profile type: Publication
full work available at URL: s00365-017-9391-5
OpenAlex ID: W2751257093
cites work: Padé approximants for functions with branch points -- strong asymptotics of Nuttall-Stahl polynomials
cites work: Q4856607
cites work: Some structural properties of two counter-examples to the Baker-Gammel-Wills conjecture.
cites work: Counter-examples to the Baker--Gammel--Wills conjecture and patchwork convergence
cites work: On rational functions without Froissart doublets
cites work: On the Baker-Gammel-Wills conjecture in the theory of Padé approximants
cites work: Convergence of the Rogers-Ramanujan continued fraction
cites work: ON CONVERGENCE OF SUBSEQUENCES OF THEmTH ROW OF A PADÉ TABLE
cites work: Multipoint Padé-type approximants. Exact rate of convergence
cites work: On sequences of rational interpolants of the exponential function with unbounded interpolation points
cites work: Simultaneous rational approximants for a pair of functions with smooth Maclaurin series coefficients
cites work: Froissart doublets in Padé approximation in the case of polynomial noise
cites work: On the phenomenon of spurious interpolation of elliptic functions by diagonal Padé approximants
cites work: Zeros of Padé error functions for functions with smooth Maclaurin coefficients
cites work: Trigonometric Padé approximants for functions with regularly decreasing Fourier coefficients
cites work: Q3222981
cites work: Rows and diagonals of the Walsh array for entire functions with smooth Maclaurin series coefficients
cites work: Pade Tables of a Class of Entire Functions
cites work: Padé tables of entire functions of very slow and smooth growth
cites work: Padé tables of entire functions of very slow and smooth growth. II
cites work: Distribution of poles of diagonal rational approximants to functions of fast rational approximability
cites work: Q5289125
cites work: Rogers-Ramanujan and the Baker-Gammel-Wills (Padé) conjecture
cites work: Exact interpolation, spurious poles, and uniform convergence of multipoint Padé approximants
cites work: ON THE CONVERGENCE OF PADÉ APPROXIMANTS IN CLASSES OF HOLOMORPHlC FUNCTIONS
cites work: Q4463727
cites work: Spurious poles in Padé approximation
cites work: Distribution of the zeros of Padé polynomials and analytic continuation
cites work: The convergence of padé approximants and the size of the power series coefficients
cites work: Riemann-Hilbert analysis and uniform convergence of rational interpolants to the exponential function
DOI: 10.1007/S00365-017-9391-5
description: scientific article; zbMATH DE number 6999349