Perturbations on the antidiagonals of Hankel matrices

Abstract

Given a strongly regular Hankel matrix, and its associated sequence of moments which defines a quasi-definite moment linear functional, we study the perturbation of a fixed moment, i.e., a perturbation of one antidiagonal of the Hankel matrix. We define a linear functional whose action results in such a perturbation and establish necessary and sufficient conditions in order to preserve the quasi-definite character. A relation between the corresponding sequences of orthogonal polynomials is obtained, as well as the asymptotic behavior of their zeros. We also study the invariance of the Laguerre-Hahn class of linear functionals under such perturbation, and determine its relation with the so-called canonical linear spectral transformations.

Introduction

Given a sequence of complex numbers ${\{{\mu }_n\}}_{n⩾0}$, one can define a linear functional $\mathcal{M}$ in the linear space of polynomials with complex coefficients $\mathbb{P}$ such that

$$〈\mathcal{M}\text{,}{x}^n〉={\mu }_n\text{.}$$

In the literature (see [6], [10], among others), $\mathcal{M}$ is said to be a moment linear functional, and the complex numbers ${\{{\mu }_n\}}_{n⩾0}$ are called the moments associated with $\mathcal{M}$. The semi-infinite matrix

$$\mathbf{H}={\left[〈\mathcal{M}\text{,}{x}^{i+j}〉\right]}_{i\text{,}j=0\text{,}1\text{,}\dots }={\left[{\mu }_{i+j}\right]}_{i\text{,}j=0\text{,}1\text{,}\dots }=\left[\begin{array}{ccccc}{\mu }_0\hfill & {\mu }_1\hfill & \cdots \hfill & {\mu }_n\hfill & \cdots \hfill \\ {\mu }_1\hfill & {\mu }_2\hfill & \cdots \hfill & {\mu }_{n+1}\hfill & \cdots \hfill \\ ⋮\hfill & ⋮\hfill & \ddots \hfill & ⋮\hfill & \cdots \hfill \\ {\mu }_n\hfill & {\mu }_{n+1}\hfill & \cdots \hfill & {\mu }_{2n}\hfill & \cdots \hfill \\ ⋮\hfill & ⋮\hfill & \ddots \hfill & ⋮\hfill & \ddots \hfill \end{array}\right]$$

is the Gram matrix associated with the bilinear form of the linear functional (1) in terms of the canonical basis ${\{x^n\}}_{n⩾0}$ of $\mathbb{P}$. If there exist a family of monic polynomials such that $\mathrm{deg}(P_n)=n$ and

$$〈\mathcal{M}\text{,}{P}_n(x)P_m(x)〉={\gamma }_n^{-2}{\delta }_{n\text{,}m}\text{,}{\gamma }_n\ne 0\text{,}n\text{,}m⩾0\text{,}$$

where ${\delta }_{n\text{,}m}$ is the Kronecker delta, then ${\{P_n\}}_{n⩾0}$ is called the monic orthogonal polynomials sequence (MOPS) associated with $\mathcal{M}$.

The Hankel matrices and their determinants play an important role in the study of moment functionals. The linear functional (1) is called quasi-definite if the moments matrix is strongly regular or, equivalently, the determinants of the principal leading submatrices ${\mathbf{H}}_n$ of order $(n+1)\times (n+1)$ are all different from 0. In this case, there exists a unique MOPS associated with $\mathcal{M}$.

On the other hand, a linear functional $\mathcal{M}$ is called positive definite if and only if its moments are all real and $\det {\mathbf{H}}_n>0\text{,}n⩾0$. In such a case, there exist a unique sequence of real polynomials ${\{p_n\}}_{n⩾0}$ orthonormal with respect to $\mathcal{M}$, i.e., the following condition holds

$$〈\mathcal{M}\text{,}{p}_n(x)p_m(x)〉={\delta }_{n\text{,}m}\text{,}$$

where

$$p_n(x)={\gamma }_n{x}^n+{\delta }_n{x}^{n-1}+(\text{lower degree terms})\text{,}{\gamma }_n>0\text{,}n⩾0\text{.}$$

From the Riesz representation theorem, we know that every positive definite linear functional $\mathcal{M}$ has an integral representation (not necessarily unique)

$$〈\mathcal{M}\text{,}{x}^n〉={\int }_I{x}^{n}d\mu (x)\text{,}$$

where μ denotes a nontrivial measure supported on some infinite subset I of the real line.

One of the most important characteristics of orthonormal polynomials on the real line is the fact that any three consecutive polynomials are connected by the simple recurrence relation

$${\mathit{xp}}_n(x)=a_{n+1}{p}_{n+1}(x)+b_{n+1}{p}_n(x)+a_n{p}_{n-1}(x)\text{,}n⩾0\text{,}$$

with initial conditions $p_{-1}\equiv 0\text{,}{p}_0\equiv {\mu }_0^{-1/2}$, and recurrence coefficients given by

$$\begin{array}{cc}\hfill & a_n={\int }_I{\mathit{xp}}_{n-1}(x)p_n(x)d\mu (x)=\frac{{\gamma }_{n-1}}{{\gamma }_n}>0\text{,}\hfill \\ \hfill & b_n={\int }_I{\mathit{xp}}_n^2(x)d\mu (x)=\frac{{\delta }_n}{{\gamma }_n}-\frac{{\delta }_{n+1}}{{\gamma }_{n+1}}\text{.}\hfill \end{array}$$

There are explicit formulae for orthonormal polynomials in terms of the determinants of the corresponding Hankel matrix. The n-th degree orthonormal polynomial is given by the Heine’s formula

$$p_n(x)=\frac{1}{\sqrt{\det {\mathbf{H}}_n\det {\mathbf{H}}_{n-1}}}\left|\begin{array}{ccccc}{\mu }_0\hfill & {\mu }_1\hfill & {\mu }_2\hfill & \dots \hfill & {\mu }_n\hfill \\ {\mu }_1\hfill & {\mu }_2\hfill & {\mu }_3\hfill & \dots \hfill & {\mu }_{n+1}\hfill \\ ⋮\hfill & ⋮\hfill & \dots \hfill & ⋮\hfill & ⋮\hfill \\ {\mu }_{n-1}\hfill & {\mu }_n\hfill & {\mu }_{n+1}\hfill & \dots \hfill & {\mu }_{2n-1}\hfill \\ 1\hfill & x\hfill & x^2\hfill & \dots \hfill & x^n\hfill \end{array}\right|\text{,}$$

while its leading coefficient is given by a ratio of two Hankel determinants

$${\gamma }_n=\sqrt{\frac{\det {\mathbf{H}}_{n-1}}{\det {\mathbf{H}}_n}}\text{.}$$

The n-th order reproducing kernel associated with ${\{p_n\}}_{n⩾0}$ is defined by

$$K_n(x\text{,}y)={\displaystyle \sum _{k=0}^n}{p}_k(x)p_k(y)\text{,}n⩾0\text{.}$$

The name comes from the fact that, for any polynomial $q_n$ of degree at most n, we have

$$q_n(y)={\int }_I{q}_n(x)K_n(x\text{,}y)d\mu (x)\text{.}$$

The reproducing kernel can be represented in a simple way in terms of the polynomials $p_n$ and $p_{n+1}$ using the Christoffel-Darboux formula (see [6], [10], among others)

$$K_n(x\text{,}y)=a_{n+1}\frac{p_{n+1}(x)p_n(y)-p_n(x)p_{n+1}(y)}{x-y}\text{,}x\ne y\text{,}$$

which can be deduced in a straightforward way from the three-term recurrence relation (3). We will denote by $K_n^{(i\text{,}j)}(x\text{,}y)$ the i-th (resp. j-th) partial derivative of $K_n(x\text{,}y)$ with respect to the variable x (resp. y). For the quasi-definite case, the reproducing kernel is defined as

$$K_n(x\text{,}y)={\displaystyle \sum _{k=0}^n}\frac{P_k(x)P_k(y)}{{\gamma }_k^2}\text{.}$$

When polynomials are studied, one of the most important quantities to be considered are their zeros. The fundamental theorem of algebra asserts that any polynomial of degree n has exactly n zeros (counting multiplicities). When dealing with orthogonal polynomials on the real line, one can say much more about their localization. Two of the most relevant properties of their zeros are the following:

• The zeros of $p_n$ are all real, simple and lie in the convex hull of I.

• Suppose $x_{n\text{,}1}<x_{n\text{,}2}<\cdots <x_{n\text{,}n}$ are the zeros of $p_n$, then

  $$x_{n\text{,}k}<x_{n-1\text{,}k}<x_{n\text{,}k+1}\text{,}1⩽k⩽n-1\text{.}$$

Our interest in perturbations of Hankel matrices is motivated by their many applications in mathematical and physical problems. Their relations with moment problems ([5]), integrable systems ([16]), Padé approximation ([13]), as well as their applications in coding theory and combinatorics (see [11], [15] and references therein), constitute an illustrative sample of their impact.

Before introducing the problem to be analyzed in this contribution, let us briefly discuss two rather straightforward but interesting examples where the moments are modified in a natural way. First, instead of taking the canonical basis of $\mathbb{P}$, consider the basis $\{1\text{,}(x-a)\text{,}{(x-a)}^2\text{,}\dots \}$, where $a\in \mathbb{R}$. Then, the new sequence of moments ${\{{\upsilon }_n\}}_{n⩾0}$ is given by

$${\upsilon }_n=〈\mathcal{M}\text{,}{(x-a)}^n〉=〈\mathcal{M}\text{,}{\displaystyle \sum _{j=0}^n}\left(\begin{array}{c}\hfill n\hfill \\ \hfill j\hfill \end{array}\right){(-1)}^{n-j}{a}^{n-j}{x}^j〉={\displaystyle \sum _{j=0}^n}\left(\begin{array}{c}\hfill n\hfill \\ \hfill j\hfill \end{array}\right){(-1)}^{n-j}{a}^{n-j}{\mu }_j\text{.}$$

As a consequence, the $(n+1)\times (n+1)$ principal leading submatrix of the corresponding Hankel matrix is

$${\stackrel{\sim }{\mathbf{H}}}_n={[{\upsilon }_{i+j-2}]}_{1⩽i\text{,}j⩽n}=\left[\begin{array}{cccc}\hfill {\mu }_0\hfill & \hfill {\mu }_1+m_1\hfill & \hfill \cdots \hfill & \hfill {\mu }_n+m_n\hfill \\ \hfill {\mu }_1+m_1\hfill & \hfill {\mu }_2+m_2\hfill & \hfill \cdots \hfill & \hfill {\mu }_{n+1}+m_{n+1}\hfill \\ \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill ⋮\hfill \\ \hfill {\mu }_n+m_n\hfill & \hfill {\mu }_{n+1}+m_{n+1}\hfill & \hfill \cdots \hfill & \hfill {\mu }_{2n}+m_{2n}\hfill \end{array}\right]\text{,}$$

where

$$m_n={\displaystyle \sum _{j=1}^{n-1}}\left(\begin{array}{c}\hfill n\hfill \\ \hfill j\hfill \end{array}\right){(-1)}^{n-j}{a}^{n-j}{\mu }_j\text{.}$$

Thus, if $\mathcal{M}$ is a quasi-definite moment linear functional, then the polynomials

$$Q_n(x)=\frac{1}{\det {\stackrel{\sim }{\mathbf{H}}}_{n-1}}\left|\begin{array}{ccccc}\hfill {\upsilon }_0\hfill & \hfill {\upsilon }_1\hfill & \hfill {\upsilon }_2\hfill & \hfill \cdots \hfill & \hfill {\upsilon }_n\hfill \\ \hfill {\upsilon }_1\hfill & \hfill {\upsilon }_2\hfill & \hfill {\upsilon }_3\hfill & \hfill \cdots \hfill & \hfill {\upsilon }_{n+1}\hfill \\ \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill \cdots \hfill & \hfill ⋮\hfill & \hfill ⋮\hfill \\ \hfill {\upsilon }_{n-1}\hfill & \hfill {\upsilon }_n\hfill & \hfill {\upsilon }_{n+1}\hfill & \hfill \cdots \hfill & \hfill {\upsilon }_{2n-1}\hfill \\ \hfill 1\hfill & \hfill (x-a)\hfill & \hfill {(x-a)}^2\hfill & \hfill \cdots \hfill & \hfill {(x-a)}^n\hfill \end{array}\right|\text{,}n⩾0\text{,}$$

constitute a sequence of monic polynomials orthogonal with respect to $\mathcal{M}$, using the new basis, with $Q_n(x-a)=P_n(x)$. Notice that this simple change of the basis resulted in a perturbation on the antidiagonals of (2). Namely, each of the $(j+1)$-th antidiagonal is perturbed by the addition of the constant $m_j$. In the remaining of the manuscript, we will use the basis $\{1\text{,}(x-a)\text{,}{(x-a)}^2\text{,}\dots \}$, since most of the required calculations can be performed in a simpler way.

The second example is given by the Uvarov’s spectral transformation (see [14], [17]), whose action results in a perturbation of the first moment ${\upsilon }_0$, while leaving the others unaffected. This perturbation (the most simple case that we can consider) is closely related with the Uvarov-Chihara integrable system (see [16]). In order to define it, we introduce the real Dirac delta functional $\delta (x-a)$ supported at $x=a$, which acts in the following way

$$〈\delta (x-a)\text{,}P(x)〉=P(a)\text{,}P\in \mathbb{P}\text{.}$$

Then, Uvarov’s transformation is defined by

$$〈{\mathcal{M}}_U\text{,}p(x)〉=〈\mathcal{M}\text{,}p(x)〉+m〈\delta (x-a)\text{,}p(x)〉=〈\mathcal{M}\text{,}p(x)〉+\mathit{mp}(a)\text{,}$$

i.e., a perturbation on the first antidiagonal on the Hankel matrix.

Now a natural question arises: Is there a linear functional $\widehat{\mathcal{M}}$ such that its action results on a perturbation of (only) the moment ${\upsilon }_j$ or, equivalently, the $(j+1)$-th antidiagonal of the Hankel matrix $\stackrel{\sim }{\mathbf{H}}$? In other words, we are interested in the properties of a functional $\widehat{\mathcal{M}}$ whose moments are given by

$${\stackrel{\sim }{\upsilon }}_n=〈\widehat{\mathcal{M}}\text{,}{(x-a)}^n〉=\left\{\begin{array}{cc}{\upsilon }_n\text{,}\hfill & n\ne j\text{,}\hfill \\ {\upsilon }_n+m_j\text{,}\hfill & n=j\text{,}\hfill \end{array}\right.$$

for some $m\in \mathbb{R}$, i.e., its corresponding Hankel matrix is

$$\mathbf{H}(m_j)=\left[\begin{array}{ccccc}\hfill {\upsilon }_0\hfill & \hfill \cdots \hfill & \hfill {\upsilon }_j+m_j\hfill & \hfill {\upsilon }_{j+1}\hfill & \hfill \cdots \hfill \\ \hfill ⋮\hfill & \hfill ⋰\hfill & \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill \cdots \hfill \\ \hfill {\upsilon }_j+m_j\hfill & \hfill \cdots \hfill & \hfill {\upsilon }_{2j}\hfill & \hfill {\upsilon }_{2j+1}\hfill & \hfill \cdots \hfill \\ \hfill {\upsilon }_{j+1}\hfill & \hfill \cdots \hfill & \hfill {\upsilon }_{2j+1}\hfill & \hfill {\upsilon }_{2j+2}\hfill & \hfill \cdots \hfill \\ \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill \ddots \hfill \end{array}\right]\text{.}$$

An analogous problem for linear functionals defined in the linear space of Laurent polynomials has been analyzed recently in [4]. There, the corresponding moments matrix is a Toeplitz matrix and the perturbation studied is one that modifies two symmetric subdiagonals of such matrix by means of a modification of the Lebesgue measure, supported on the unit circle. The main objective of this manuscript is to study equivalent perturbations of Hankel matrices, which appear in the theory of orthogonal polynomials on the real line. In Section 2, a linear functional with the desired properties, denoted by ${\mathcal{M}}_j$, is defined in terms of $\mathcal{M}$, and we obtain necessary and sufficient conditions for the quasi-definiteness of ${\mathcal{M}}_j$, provided that $\mathcal{M}$ is quasi-definite. Under those conditions, we obtain an expression that relates their corresponding families of orthogonal polynomials. In Section 3, we analyze the connection between ${\mathcal{M}}_j$ and the so-called canonical spectral transformations, and the invariance of the Laguerre Hahn class under this perturbation is studied in Section 4. Section 5 deals with an asymptotic analysis of the zeros of the corresponding orthogonal polynomials. Finally, in Section 6, we pose some related open problems that will be considered in future contributions.