| Russian Mathematical Surveys |
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This article is cited in 3 scientific papers (total in 3 papers) Brief communications Convergence of Hermite–Padé rational approximations S. P. Suetin Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
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     1.Let $f$ be a function defined by a power series at the point $z=\infty$: Given the first $N$ coefficients $c_0,\dots,c_{N-1}$ of the series in (1), we can construct from them a few constructive approximations to the original function $f$ (see [10]; here we mean ‘constructive’ in the sense specified in [2], § 2). It is natural to wonder if such approximations are optimal. In what follows we assume that $N=2n+1=3m+1=4\ell+1$, где $n,m,\ell\in\mathbb{N}$. This is not a substantative condition and has a purely technical nature (see (2)–(4)).The best known, popular, and well investigated constructive approximations to a power series are the Padé approximations $[n/n]_f=P_n/Q_n$, where the polynomials $P_n,Q_n\not\equiv0$, $\deg{P_n},\deg{Q_n}\leqslant n$, are specified by the relation The convergence of Padé approximations in the class of multivalued analytic functions with a finite number of singular points follows from Stahl’s theory of 1985–1986 [7]. On the other hand there also exist some other methods, which are not as well known, for approximating a power series (1) given the set of its first $N$ coefficients. Here we discuss rational approximations constructed on the basis of Hermite–Padé polynomials of the second type for the systems $f$, $f^2$ and $f$, $f^2$, $f^3$. It turns out that, from the standpoint of the optimal use of the $N$ coefficients of the series (1), Hermite–Padé rational approximations have certain advantages over Padé approximations (see (10); also see [4] and [3]). Their convergence has so far been considered only in special cases (see [1], [6], [5], and [4]). The paper [3] presents some numerical results related to van der Pol’s equation. They show that the use of Hermite–Padé polynomials for the systems $f$, $f^2$ and $f$, $f^2$, $f^3$ extends significantly one’s abilities to analyse the properties of the function $f$ numerically on the basis of the coefficients of the series (1). In particular, one can distinguish in this way quadratic branch points of a function given by a series (see [9]). The aim of this note is to present theoretical results showing that Hermite–Pade rational approximations are more optimal than Padé approximations.For the system $f$, $f^2$ we define the Hermite–Padé polynomials $P^{(2)}_{2m,0}\not\equiv 0$, $P^{(2)}_{2m,1}$, and $P^{(2)}_{2m,2}$, $\deg{P^{(2)}_{2m,j}}\leqslant 2m$, by the relations
$$
\begin{equation}
\begin{alignedat}{2} (P^{(2)}_{2m,0}f-P^{(2)}_{2m,1})(z)&=O(z^{-m-1}),&\qquad z&\to\infty, \\ (P^{(2)}_{2m,0}f^2-P^{(2)}_{2m,2})(z)&=O(z^{-m-1}),&\qquad z&\to\infty. \end{alignedat}
\end{equation}
\tag{3}
$$
In a similar way, for the system $f$, $f^2$, $f^3$ we define the Hermite–Padé polynomials $P^{(3)}_{3\ell,0}\not\equiv0$, $P^{(3)}_{3\ell,1}$, $P^{(3)}_{3\ell,2}$, and $P^{(3)}_{3\ell,3}$, $\deg{P^{(3)}_{3\ell,j}}\leqslant 3\ell$, by the relations
$$
\begin{equation}
\begin{alignedat}{2} (P^{(3)}_{3\ell,0}f-P^{(3)}_{3\ell,1})(z)&=O(z^{-\ell-1}),&\qquad z&\to\infty, \\ (P^{(3)}_{3\ell,0}f^2-P^{(3)}_{3\ell,2})(z)&=O(z^{-\ell-1}),&\qquad z&\to\infty, \\ (P^{(3)}_{3\ell,0}f^3-P^{(3)}_{3\ell,3})(z)&=O(z^{-\ell-1}),&\qquad z&\to\infty. \end{alignedat}
\end{equation}
\tag{4}
$$
2.Let $f\in\mathcal{H}(\infty)$ be explicitly defined by $f(z):=[(A-1/{\varphi(z)})\times(B-1/{\varphi(z)})]^{-1/2}$, $z\notin E:=[-1,1]$, where $1<A<B$, $\varphi(z)=z+(z^2-1)^{1/2}$ and the branch of $(\,\cdot\,)^{1/2}$ is selected so that $\varphi(z)\sim 2z$ as $z\to\infty$. We denote the class of these functions by $\mathcal Z(E)$. The function $f$ is algebraic of order four, with four branch points $\pm1$, $a$, and $b$, where $a=(A+1/A)/2$ and $b=(B+1/B)/2$, $1<a<b$. All branch points of $f$ are of the second order. It was shown in [8] that $f$, $f^2$, $f^3$ is a Nikishin system:
$$
\begin{equation}
\begin{gathered} \, f(z)=(AB)^{-1/2}+\widehat{\sigma}(z),\qquad f^2(z)=(AB)^{-1}+(AB)^{-1/2}\,\widehat{\sigma}(z)+\widehat{s}_1(z), \\ f^3(z)=(AB)^{-3/2}+(AB)^{-1}\,\widehat{\sigma}(z)+(AB)^{-1/2}\,\widehat{s}_1(z)+ \widehat{s}_2(z), \end{gathered}
\end{equation}
\tag{5}
$$
where $\operatorname{supp}\sigma=\operatorname{supp}{s}_1= \operatorname{supp}{s}_2=E$, $\operatorname{supp}{\sigma}_2=F:=[a,b]$, $s_1:=\langle\sigma,\sigma_2\rangle$, and $s_2:=\langle\sigma,\sigma_2,\sigma\rangle$. Let $M_1(F)$ denote the class of unit (Borel) measures with support on $F$. Let $g_E(t,z)$ be the Green’s function for the domain $D:=\widehat{\mathbb{C}}\setminus{E}$ with singularity at $t=z$, and let $V^\mu(z)$ be the logarithmic potential and $G^\mu_E(z)$ be the Green’s potential of $\mu\in M_1(F)$:
$$
\begin{equation}
V^\mu(z):=\int\log\frac{1}{|t-z|}\,d\mu(t)\quad\text{and}\quad G^\mu_E(z):=\int g_E(t,z)\,d\mu(t),\quad z\in\widehat{\mathbb{C}}\setminus{F}.
\end{equation}
\tag{6}
$$
For each $\theta\in(0,\infty)$ there exists (see [6]) a unique measure $\lambda=\lambda(\theta)\in M_1(F)$ such that the equilibrium condition $\theta V^\lambda(x)+G^\lambda_E(x)+ \theta g_E(x,\infty)\equiv\operatorname{const}$, $x\in F$, is satisfied. We have the following result. Theorem 1. Let $f\in\mathcal Z(E)$. Then as $N\to\infty$, locally uniformly in $D$, It follows from Stahl’s theorem that
$$
\begin{equation}
\lim_{N\to\infty}|f(z)-[n/n]_f(z)|^{1/N}=\exp\{-g_E(z,\infty)\}=:\delta_1(z).
\end{equation}
\tag{9}
$$
We can show that $G^{\lambda(1)}_E(z)>2G^{\lambda(3)}_E(z)/3$, $z\in D$. From (9), (7), and (8) we obtain
Citation in format AMSBIB
\Bibitem{Sue23} |
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