On uniform convergence of diagonal multipoint Padé approximants for entire functions
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 \(f\) given by \((2)\), \[ \limsup_{j\rightarrow\infty}\,|a_j|^{1/j^2}<\frac{1}{3}. \] Then there is a subsequence \(\mathcal S\) of integers with the property \((2)\) as described in Theorem 1.1. Based on the result in Theorem 1.1, the author poses a stronger \textit{Baker-Gammel-Wills conjecture} for entire functions: Let \(f\) be entire. Then there is an infinite subsequence \(\mathcal S\) of positive integers with the following property: given any \(r>0\) and for \(n\in{\mathcal S}\), multipoint Padé approximants \(R_n\) to \(f\) of type \((n,n)\) formed for interpolation points \(\Lambda_n\subset{\mathcal B}_r\), we have \((1)\).
scientific article; zbMATH DE number 6999349
scientific article