| Russian Mathematical Surveys |
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This article is cited in 4 scientific papers (total in 4 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=∞: Given the first N coefficients c0,…,cN−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=4ℓ+1, где n,m,ℓ∈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=Pn/Qn, where the polynomials Pn,Qn≢0, degPn,degQn⩽, 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}
This publication is cited in the following 4 articles:
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