The linear pencil approach to rational interpolation

TL;DR

This paper extends the theory of rational interpolation by analyzing linear pencils of matrices, providing new convergence criteria, spectral analysis, and links to biorthogonal rational functions, broadening classical results in Pade approximation.

Contribution

It introduces a new criterion for the resolvent set of linear pencils and generalizes convergence results for rational interpolants beyond classical Pade approximants.

Findings

New resolvent set criterion for linear pencils

Generalized convergence results for rational interpolants

Explicit spectral and numerical range calculations for specific cases

Abstract

It is possible to generalize the fruitful interaction between (real or complex) Jacobi matrices, orthogonal polynomials and Pade approximants at infinity by considering rational interpolants, (bi-)orthogonal rational functions and linear pencils zB-A of two tridiagonal matrices A, B, following Spiridonov and Zhedanov. In the present paper, beside revisiting the underlying generalized Favard theorem, we suggest a new criterion for the resolvent set of this linear pencil in terms of the underlying associated rational functions. This enables us to generalize several convergence results for Pade approximants in terms of complex Jacobi matrices to the more general case of convergence of rational interpolants in terms of the linear pencil. We also study generalizations of the Darboux transformations and the link to biorthogonal rational functions. Finally, for a Markov function and for pairwise conjugate interpolation points tending to infinity, we compute explicitly the spectrum and the numerical range of the underlying linear pencil.

Equations

$$\phi (z)={\int }_a^b\frac{d\mu (t)}{z-t},$$

$$\phi (z)=\frac{1}{z-b_0-\frac{a_0^2}{z-b_1-\frac{a_1^2}{\ddots }}}=\text{lower}8.94444pt\text{vrule}height=11.44444pt,depth=1.80444pt\frac{1}{z-b_0}\text{raise}2.5pt\text{vrule}height=10.94444pt-\text{lower}8.94444pt\text{vrule}height=11.44444pt,depth=1.80444pt\frac{a_0^2}{z-b_1}\text{raise}2.5pt\text{vrule}height=12.41443pt-\text{lower}8.94444pt\text{vrule}height=11.44444pt,depth=1.80444pt\frac{a_1^2}{z-b_2}\text{raise}2.5pt\text{vrule}height=12.41443pt-\cdots ,$$

$$A=\left(\begin{array}[]{cccc}b_{0}&a_{0}&&\\ a_{0}&b_{1}&a_{1}&\\ &a_{1}&b_{2}&\ddots\\ &&\ddots&\ddots\\ \end{array}\right),\quad A_{[0:n-1]}=\left(\begin{array}[]{cccc}b_{0}&a_{0}&&\\ a_{0}&b_{1}&\ddots&\\ &\ddots&\ddots&a_{n-2}\\ &&a_{n-2}&b_{n-1}\\ \end{array}\right).$$

$$\frac{p_n(z)}{q_n(z)}=⟨(zI-A_{[0:n-1]}{)}^{-1}{e}_0,e_0⟩=\text{lower}6.30554pt\text{vrule}height=8.80554pt,depth=1.80444pt\frac{1}{z-a_0}\text{raise}2.5pt\text{vrule}height=10.94444pt-\cdots -\text{lower}6.30554pt\text{vrule}height=8.80554pt,depth=1.80444pt\frac{b_{n-2}^2}{z-a_{n-1}}\text{raise}2.5pt\text{vrule}height=15.05333pt,$$

$$\phi (z)-\frac{p_n(z)}{q_n(z)}=O{\left(\frac{1}{z^{2n+1}}\right)}_{z\to \infty },$$

$$\text{lower}8.94444pt\text{vrule}height=11.44444pt,depth=1.80444pt\frac{1}{z-b_0}\text{raise}2.5pt\text{vrule}height=10.94444pt-\text{lower}8.94444pt\text{vrule}height=11.44444pt,depth=1.80444pt\frac{a_0^2(z-z_1)(z-z_2)}{z-b_1}\text{raise}2.5pt\text{vrule}height=14.5pt-\text{lower}8.94444pt\text{vrule}height=11.44444pt,depth=1.80444pt\frac{a_1^2(z-z_3)(z-z_4)}{z-b_2}\text{raise}2.5pt\text{vrule}height=14.5pt-\dots ,$$

$$⟨x,y⟩=\sum _{j=0}^{\infty }{x}_j{\overline{y}}_j,x,y\in {\ell }^2.$$

$$\text{lower}9.5pt\text{vrule}height=12.0pt,depth=2.5pt\frac{1}{{\beta }_0(z)}\text{raise}2.5pt\text{vrule}height=10.94444pt-\text{lower}9.5pt\text{vrule}height=12.0pt,depth=2.5pt\frac{{\alpha }_0^L(z){\alpha }_0^R(z)}{{\beta }_1(z)}\text{raise}2.5pt\text{vrule}height=14.5pt-\text{lower}9.5pt\text{vrule}height=12.0pt,depth=2.5pt\frac{{\alpha }_1^L(z){\alpha }_1^R(z)}{{\beta }_2(z)}\text{raise}2.5pt\text{vrule}height=14.5pt-\dots$$

$$y_{n+1}={\beta }_n(z)y_n-{\alpha }_{n-1}^L(z){\alpha }_{n-1}^R(z)y_{n-1},n=0,1,2,\dots$$

$$q_0(z)=1,q_{-1}(z)=0,p_0(z)=0,p_{-1}(z)=-1.$$

$$q_n(z)=\det (z{B}_{[0:n-1]}-A_{[0:n-1]}),p_n(z)=\det (z{B}_{[1:n-1]}-A_{[1:n-1]}).$$

$$C_n(z)=\frac{p_n(z)}{q_n(z)}=⟨(z{B}_{[0:n-1]}-A_{[0:n-1]}{)}^{-1}{e}_0,e_0⟩.$$

$$p_{n+1}(z)q_n(z)-p_n(z)q_{n+1}(z)=\prod _{k=0}^{n-1}{\alpha }_k^L(z){\alpha }_k^R(z),n=0,1,2,\dots .$$

$$r_n(z):=\varphi (z)q_n(z)-p_n(z),$$

$$r_0(z)=\varphi (z),r_{-1}(z)=1.$$

$$zB-A=\left(\begin{array}{ccccc}{\beta }_0(z)& -{\alpha }_0^R(z)& 0& 0& \dots \\ -{\alpha }_0^L(z)& {\beta }_1(z)& -{\alpha }_1^R(z)& 0& \ddots \\ 0& -{\alpha }_1^L(z)& {\beta }_2(z)& {\alpha }_2^R(z)& \ddots \\ ⋮& \ddots & \ddots & \ddots & \ddots \end{array}\right)$$

$$A{q}^R(z)=zB{q}^R(z),q^L(z)A=z{q}^L(z)B,$$

$$q_n^R(z)=\frac{q_n(z)}{\prod _{k=0}^{n-1}{\alpha }_k^R(z)},p_n^R(z)=\frac{p_n(z)}{\prod _{k=0}^{n-1}{\alpha }_k^R(z)},r_n^R(z)=\frac{r_n(z)}{\prod _{k=0}^{n-1}{\alpha }_k^R(z)},$$

$${\alpha }_n^R(z)y_{n+1}^R-{\beta }_n(z)y_n^R+{\alpha }_{n-1}^L(z)y_{n-1}^R=0,n=0,1,2,\dots .$$

$$q_n^L(z)=\frac{q_n(z)}{\prod _{k=0}^{n-1}{\alpha }_k^L(z)},p_n^L(z)=\frac{p_n(z)}{\prod _{k=0}^{n-1}{\alpha }_k^L(z)},r_n^L(z)=\frac{r_n(z)}{\prod _{k=0}^{n-1}{\alpha }_k^L(z)},$$

$${\alpha }_n^L(z)y_{n+1}^L-{\beta }_n(z)y_n^L+{\alpha }_{n-1}^R(z)y_{n-1}^L=0,n=0,1,2,\dots .$$

$$p^L(z)(zB-A)=-e_0^{\top },(zB-A)p^R(z)=-e_0.$$

$$D_{1}z-D_2=\text{mbox}diag{\left(\frac{z-z_n}{1+|z_n|}\right)}_{n=0,1,2,...},$$

$$Bz-A=U^{*}(D_{1}z-D_{2})U,\qquad U=\left(\begin{array}[]{cccccc}1&-1/2&0&\cdots&\cdots\\ 0&1&-1/2&0&\cdots\\ 0&0&1&-1/2&\ddots\\ \vdots&\ddots&\ddots&\ddots&\ddots\end{array}\right)$$

$${\alpha }_n^L(z)={\alpha }_n^R(z)=\frac{1}{2}\frac{z-z_n}{1+|z_n|},q_n^L(z)=q_n^R(z)=2^n.$$

$$R(z)_{j,k}=\left\{\begin{array}[]{cc}r_{j}^{R}(z)q_{k}^{L}(z)=(q_{j}^{R}(z)\phi(z)-p_{j}^{R}(z))q_{k}^{L}(z)&\mbox{if $j\geq k$,}\\ q_{j}^{R}(z)r_{k}^{L}(z)=q_{j}^{R}(z)(q_{k}^{L}(z)\phi(z)-p_{k}^{L}(z))&\mbox{if $j\leq k$.}\end{array}\right.$$

$$|R(z{)}_{j,k}|\le \gamma (z)\delta (z{)}^{|j-k|},j,k=0,1,....$$

$$\varphi (z)=R(z{)}_{0,0}=⟨(zB-A{)}^{-1}{e}_0,e_0⟩.$$

$$m(z)=⟨(zB-A{)}^{-1}{e}_0,e_0⟩,z\in \rho (A,B)$$

$$\underset{n\to \infty }{\mathrm{limsup}}|r_n^L(z){|}^{1/n}<1,\underset{n\to \infty }{\mathrm{limsup}}|r_n^R(z){|}^{1/n}<1,z\in \rho (A,B).$$

Full text

In the present paper, beside revisiting the underlying generalized Favard theorem, we suggest a new criterion for the resolvent set of this linear pencil in terms of the underlying associated rational functions. This enables us to generalize several convergence results for Padé approximants in terms of complex Jacobi matrices to the more general case of convergence of rational interpolants in terms of the linear pencil. We also study generalizations of the Darboux transformations and the link to biorthogonal rational functions. Finally, for a Markov function and for pairwise conjugate interpolation points tending to $\infty$, we compute explicitly the spectrum and the numerical range of the underlying linear pencil.

1. Introduction

The connection with Jacobi matrices has led to numerous applications of spectral techniques for self-adjoint operators in the theory of orthogonal polynomials on the real line and Padé approximation. In order to give an idea of these interactions consider a Markov function of the form

[TABLE]

where $a,b$ are real numbers and $d\mu(t)$ is a probability measure, that is, $\int_{a}^{b}d\mu(t)=1$. It is well known [1], [33] that one can expand such a Markov function $\varphi$ into the following continued fraction

[TABLE]

where $b_{j},a_{j}\in\mathbb{R}$, $a_{j}>0$. Continued fractions of the form (1.1) are called J-fractions [23, 33]. To the continued fraction (1.1) one can associate a Jacobi matrix $A$ acting in the space of square summable sequences and its truncation $A_{[0:n]}$

[TABLE]

Then it is known that $\varphi(z)=\langle(zI-A)^{-1}e_{0},e_{0}\rangle$, and the $n$th convergent of the above continued fraction is given by

[TABLE]

where the column vector $e_{0}=(1,0,\dots)^{\top}$ is the first canonical vector of suitable size, $q_{n}$ are orthogonal polynomials with respect to $d\mu$, and $p_{n}$ are polynomials of the second kind, see [1, 26, 27]. It is elementary fact of the continued fraction theory that

[TABLE]

see for instance [1, 4, 23]. Relation (1.2) means that the rational function $p_{n}/q_{n}$ is the $n$th diagonal Padé approximant to $\varphi$ at infinity. Consequently, the locally uniform convergence of diagonal Padé approximants appears as the strong resolvent convergence of the finite matrix approximations $A_{[0:n]}$. For instance, one knows that $p_{n}/q_{n}\to\varphi$ in capacity in the resolvent set of $A$ given by the complement of the support of $\mu$, and locally uniformly outside the numerical range of $A$ given by the convex hull of the spectrum of $A$, see for instance [30]. Besides, it should be mentioned here that an operator approach for proving convergence of Padé approximants for rational perturbations of Markov functions was proposed in [17], see also [16].

If $\varphi$ is no longer a Markov function but has distinct $n$th diagonal Padé approximants at infinity, we may still recover these approximants as convergents of a continued fraction of type (1.1), but now in general $a_{j},b_{j}\in\mathbb{C}$, $a_{j}\neq 0$, see [33], that is, $A$ becomes complex symmetric, called a complex Jacobi matrix. There is no longer a natural candidate for the spectrum of $A$, but it is still possible to characterize the spectrum in terms of some asymptotic behavior of the Padé denominators $q_{n}(z)$ and the linearized error functions $r_{n}(z)=q_{n}(z)\phi(z)-p_{n}(z)$ [3, 12, 10], see also [8, 17, 16] for more general banded matrices. Convergence outside the numerical range was established in [12], and convergence in capacity in the outer connected component of the resolvent set in [11]. We refer the reader to [9] for some recent summary on complex Jacobi matrices, including some open questions partially solved in [7].

The goal of this paper is to generalize several of the above results to the case of multipoint Padé approximants.

Definition 1.1** ([4]).**

The $[n_{1}|n_{2}]$ multipoint Padé approximant (or rational interpolant) for a function $\varphi$ at the points $\{z_{k}\}_{k=1}^{\infty}$ is defined as the ratio $p/q$ of two polynomials $p$ and $q\neq 0$ of degree at most $n_{1}$ and $n_{2}$, respectively, such that $\varphi q-p$ vanishes at $z_{1},z_{2},...,z_{n_{1}+n_{2}+1}$ counting multiplicities.

It is easy to see that the degree and interpolation conditions lead to a homogeneous system of linear equations, and thus an $[n_{1}|n_{2}]$ multipoint Padé approximant exists. Also, one may show uniqueness of the fraction $p/q$. However, since the denominator may vanish at some of the interpolation points, it may happen that the fraction $p/q$ does not interpolate $\varphi$ at some point $z_{k}$, usually referred to as an unattainable point.

Under some regularity conditions, the $[n-1|n]$ multipoint Padé approximants of $\varphi$ may be written as $n$th convergents of a continued fraction of the form

[TABLE]

the odd part of a Thiele continued fraction [4]. Continued fractions of this type are referred to as $MP$-fractions in [21] and as $R_{II}$–fractions in [22]. In particular, the authors study in [21, Theorem  4.4] and [22, Theorem 3.5] some analog of Favard’s theorem and the link with orthogonal rational functions. Spiridonov and one of the authors [29, 34] showed that such continued fractions are related not to a single Jacobi matrix but to a pencil $zB-A$ with tridiagonal matrices $A,B$. Various links to bi-orthogonal rational functions have been presented in [34] and [18]. In particular, in [18, Theorem 6.2] the authors present an operator-theoretic proof for the Markov convergence theorem multipoint Padé approximants [20] based on spectral properties of the pencil $zB-A$.

The aim of this paper is to present further convergence results for the continued fraction (1.3), both in the resolvent set and outside the numerical range of the tridiagonal linear pencil $zB-A$. To be more precise, denote by $\ell^{2}=\ell^{2}_{[0:\infty)}$ the Hilbert space of complex square summable sequences $(x_{0},x_{1},\dots)^{\top}$ with the usual inner product

[TABLE]

We will restrict our attention to the case of tridiagonal matrices $A,B$ with bounded entries, in which case we may identify via usual matrix product the matrices $A$ and $B$ with bounded operators acting in $\ell^{2}$. Notice that many algebraic relations remain true in the unbounded case as well. However, already the simpler case of bounded pencils allows to describe the main ideas of how to generalize results from the classical theory of orthogonal polynomials to the theory of biorthogonal rational functions as well as to the multipoint Padé approximation.

The remainder of the paper is organized as follows: we start from a general bounded $MP$–fraction and introduce the associated linear pencils together with the rational solutions of some underlying three term recurrence relations in §2.1. In §2.2, by generalizing previous work of Aptekarev, Kaliaguine & Van Assche [3] we show how the asymptotic behavior of these rational solutions allows to decide whether the linear pencil $zB-A$ is boundedly invertible. In particular, we deduce in Corollary 2.6 the pointwise convergence of at least a subsequence of our multipoint Padé approximants towards what is called the $m$–function (or Weyl function) of the linear penc