Polynomial perturbations of hermitian linear functionals and difference equations

TL;DR

This paper introduces a new method for analyzing polynomial modifications of hermitian functionals on the unit circle, characterizing quasi-definiteness via difference equations for Schur parameters, with applications to orthogonal polynomials.

Contribution

It presents a novel approach to study polynomial modifications of arbitrary degree for hermitian functionals, linking quasi-definiteness to recurrence relations of Schur parameters.

Findings

Characterization of quasi-definiteness through difference equations.

Analysis of degree one polynomial perturbations of Lebesgue measure.

Classification of orthogonal polynomials related by linear relations.

Abstract

This paper is devoted to the study of general (Laurent) polynomial modifications of moment functionals on the unit circle, i.e., associated with hermitian Toeplitz matrices. We present a new approach which allows us to study polynomial modifications of arbitrary degree. The main objective is the characterization of the quasi-definiteness of the functionals involved in the problem in terms of a difference equation relating the corresponding Schur parameters. The results are presented in the general framework of (non necessarily quasi-definite) hermitian functionals, so that the maximum number of orthogonal polynomials is characterized by the number of consistent steps of an algorithm based on the referred recurrence for the Schur parameters. Some concrete applications to the study of orthogonal polynomials on the unit circle show the effectiveness of this new approach: an exhaustive and instructive analysis of the functionals coming from a general inverse polynomial perturbation of degree one for the Lebesgue measure; the classification of those pairs of orthogonal polynomials connected by a kind of linear relation with constant polynomial coefficients; and the determination of those orthogonal polynomials whose associated ones are related to a degree one polynomial modification of the original orthogonality functional.

Figures

Equations

$$(f,g{)}_v:=v[f_{\ast }g],f,g\in \Lambda .$$

$$v[f]={\int }_{\mathbb{T}}f(z)d\mu (z),f\in \Lambda .$$

$${\psi }_n(z)=z{\psi }_{n-1}(z)+{\psi }_n(0){\psi }_{n-1}^{\ast }(z),n=1,2\dots ,$$

$${\psi }_n^{\ast }(z)=\overline{{\psi }_n(0)}z{\psi }_{n-1}(z)+{\psi }_{n-1}^{\ast }(z),n=1,2\dots .$$

$$1-|{\psi }_n(0){|}^2=\frac{{\epsilon }_n}{{\epsilon }_{n-1}},n=1,2,\dots ,$$

$$vL[f]:=v[Lf],f\in \Lambda .$$

$$A{\phi }_j=C_j{\psi }_{j+r}+D_j{\psi }_{j+r}^{\ast },j=0,\dots ,n.$$

$$A{\phi }_j^{\ast }=z{D}_j^{\ast }{\psi }_{j+r}+C_j^{\ast }{\psi }_{j+r}^{\ast },D_j^{\ast }=D_j^{{\ast }_{r-1}},j=0,\dots ,n.$$

$$0=u\bigl{[}\varphi_{j}z^{-k}\bigr{]}=\lambda v\bigl{[}A\varphi_{j}z^{-(k+r)}\bigr{]},\qquad r\leq k+r\leq j+r-1,$$

$$vL[\varphi_{j}z^{-k}\bigl{]}=v\bigl{[}A\varphi_{j}z^{-(k+r)}\bigl{]}=v\bigl{[}\bigl{(}C_{j}\psi_{j+r}+D_{j}\psi_{j+r}^{*}\bigr{)}z^{-(k+r)}\bigr{]}=0$$

$$vL\bigl{[}\varphi_{j}z^{-j}\bigl{]}=v\bigl{[}A\varphi_{j}z^{-(j+r)}\bigl{]}=v\bigl{[}\bigl{(}C_{j}\psi_{j+r}+D_{j}\psi_{j+r}^{*}\bigr{)}z^{-(j+r)}\bigr{]}=C_{j}(0)\varepsilon_{j+r}.$$

$$A=C_0{\psi }_r+D_0{\psi }_r^{\ast },C_0\in {\mathbb{P}}_r,D_0\in {\mathbb{P}}_{r-1},$$

$$A{\phi }_n=C_n{\psi }_{n+r}+D_n{\psi }_{n+r}^{\ast },n\ge 0,$$

$$A{\phi }_n^{\ast }=z{D}_n^{\ast }{\psi }_{n+r}+C_n^{\ast }{\psi }_{n+r}^{\ast },n\ge 0.$$

$$\begin{array}[]{c}a_{j}=\varphi_{j}(0),\qquad b_{j}=\psi_{j}(0),\qquad e_{j}=(\varphi_{j},\varphi_{j})_{u},\qquad\varepsilon_{j}=(\psi_{j},\psi_{j})_{v},\vskip 6.0pt plus 2.0pt minus 2.0pt\\ \Phi_{j}=\pmatrix{\varphi_{j}\cr\varphi_{j}^{*}},\qquad\mathcal{S}_{j}=\pmatrix{z&a_{j}\cr z\overline{a}_{j}&1},\qquad\mathcal{A}_{j}=\pmatrix{1&a_{j}\cr\overline{a}_{j}&1},\vskip 6.0pt plus 2.0pt minus 2.0pt\\ \Psi_{j}=\pmatrix{\psi_{j}\cr\psi_{j}^{*}},\qquad\mathcal{T}_{j}=\pmatrix{z&b_{j}\cr z\overline{b}_{j}&1},\qquad\mathcal{B}_{j}=\pmatrix{1&b_{j}\cr\overline{b}_{j}&1},\vskip 6.0pt plus 2.0pt minus 2.0pt\\ \mathcal{C}_{j}=\pmatrix{C_{j}&D_{j}\cr zD_{j}^{*}&C_{j}^{*}},\qquad\tilde{\mathcal{C}}_{j}=\pmatrix{C_{j}&zD_{j}\cr D_{j}^{*}&C_{j}^{*}}.\end{array}$$

$${\Phi }_j={\mathcal{S}}_j{\Phi }_{j-1},{\Psi }_j={\mathcal{T}}_j{\Psi }_{j-1},$$

$$A{\Phi }_j={\mathcal{C}}_j{\Psi }_{j+r}.$$

$$\mathcal{C}=\pmatrix{C&D\cr zD^{*}&C^{*}},\quad\deg C=r,\quad\deg D\leq r-1,$$

$$\tilde{\mathcal{C}}=\pmatrix{C&zD\cr D^{*}&C^{*}},$$

$$a{C}^{\ast }(0)=bC(0)+D(0).$$

$$a\overline{C(0)}=b\overline{C^{\ast }(0)}-\overline{D^{\ast }(0)}.$$

$$\hat{\mathcal{C}}=\pmatrix{z^{-1}&0\cr 0&1}\mathcal{X}\pmatrix{z&0\cr 0&1},\quad\mathcal{X}=\frac{1}{1-|a|^{2}}\pmatrix{1&-a\cr-\overline{a}&1}\mathcal{C}\pmatrix{1&b\cr\overline{b}&1}.$$

$$\hat{\mathcal{C}}=\frac{1}{1-|b|^{2}}\pmatrix{1&a\cr\overline{a}&1}\tilde{\mathcal{C}}\pmatrix{1&-b\cr-\overline{b}&1}.$$

$$M=\pmatrix{M_{1}&M_{2}\cr M_{3}&M_{4}},\qquad N=\pmatrix{N_{1}&N_{2}\cr N_{3}&N_{4}},$$

$$M\text{pmatrix}P\text{cr}Q=N\text{pmatrix}P\text{cr}Q\iff M=N.$$

$$A{\Phi }_j={\mathcal{C}}_j{\Psi }_{j+r},j=0,\dots ,n.$$

$$A{\Phi }_n={\mathcal{C}}_n{\Psi }_{n+r}.$$

$${\mathcal{C}}_0{\Psi }_r=A\text{pmatrix}1\text{cr}1,{\mathcal{C}}_0\in {\mathbb{J}}_r.$$

$${\mathcal{C}}_j{\mathcal{T}}_{j+r}={\mathcal{S}}_j{\mathcal{C}}_{j-1},j=1,\dots ,n,$$

$${\mathcal{C}}_j{\mathcal{B}}_{j+r}={\mathcal{A}}_j{\tilde{\mathcal{C}}}_{j-1},j=1,\dots ,n,$$

Full text

The main objective is the characterization of the quasi-definiteness of the functionals involved in the problem in terms of a difference equation relating the corresponding Schur parameters. The results are presented in the general framework of (non necessarily quasi-definite) hermitian functionals, so that the maximum number of orthogonal polynomials is characterized by the number of consistent steps of an algorithm based on the referred recurrence for the Schur parameters.

Some concrete applications to the study of orthogonal polynomials on the unit circle show the effectiveness of this new approach: an exhaustive and instructive analysis of the functionals coming from a general inverse polynomial perturbation of degree one for the Lebesgue measure; the classification of those pairs of orthogonal polynomials connected by a kind of linear relation with constant polynomial coefficients; and the determination of those orthogonal polynomials whose associated ones are related to a degree one polynomial modification of the original orthogonality functional.

Keywords and phrases: Orthogonal polynomials, Hermitian functionals, Schur parameters.

(2000) AMS Mathematics Subject Classification: 42C05.

1 Introduction

The intense activity during the last decades around the theory of orthogonal polynomials on the unit circle has stimulated the study of perturbations of hermitian functionals. The possibility of considering modifications that do not preserve the hermitian character of the functional leads to left and right orthogonal polynomials (see [3]), thus most of the efforts have been concentrated in the analysis of hermitian perturbations as a source of new families of standard orthogonal polynomials (see the recent monograph on orthogonal polynomials on the unit circle [22, 23]) and the references therein).

This paper proposes a new method to study the hermitian modifications obtained when multiplying a hermitian functional by a Laurent polynomial of any degree, in short, the hermitian polynomial modifications. This kind of perturbation has been considered previously (see for instance [1, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 24]), but the usual approaches have the drawback of being formulated in terms of orthogonal polynomials, kernels and determinants, what makes difficult the practical application, specially for perturbations of high degree.

On the contrary, our method is based on a recurrence for the Schur parameters of the two functionals involved in the perturbation. This provides an algorithm to generate the Schur parameters of one of the functionals, starting from the Schur parameters of the other functional. Furthermore, this recurrence yields a characterization of the maximum number of orthogonal polynomials for one of the functionals, given the number of orthogonal polynomials that the other functional has. That is, this approach permits us to study the relation between the quasi-definiteness of a functional and a polynomial modification of any degree with less computational effort than the methods already existing.

We distinguish between three different but related problems, depending on the data at hand:

• •

Basic problem: characterize when two functionals are related by a polynomial perturbation in terms of their Schur parameters.

• •

Direct problem: characterize the quasi-definiteness of a polynomial modification from the Schur parameters of the original functional.

• •

Inverse problem: characterize the quasi-definiteness of a functional from the Schur parameters of one of its polynomial modifications.

Despite the symmetry between the direct and inverse problems, they have a quite different nature which makes much more interesting the last one. The root of this difference is the fact that, given a functional and a Laurent polynomial, the corresponding polynomial modification is uniquely defined while there are infinitely many functionals whose modification is the given one. This leads to a rich structure in the set of solutions of the inverse problem which, as we will see, is related to another kind of interesting modifications: the addition of Dirac deltas and its derivatives. Hence, any information about inverse polynomial modifications can be translated as a result on perturbations by Dirac deltas.

Furthermore, as an example will show, some special solutions of an inverse problem can act as “attractors” for the asymptotics of the parameters of other solutions. Thus, the analysis of those special solutions provides information about the asymptotics of perturbations by Dirac deltas.

The rich structure of the inverse problem has a double interest due to the fact that our approach, based on a recurrence for the Schur parameters, also yields interesting connections between the study of polynomial modifications and difference equations. Therefore, the asymptotics of the solutions of the inverse problem is closely related to the asymptotics of difference equations.

The content of the paper is structured in the following way: the rest of the introduction summarizes the basic definitions and notations; Section 2 includes the main results about hermitian polynomial modifications, i.e., it is devoted to what we call basic problem; direct and inverse problems are discussed in Section 3, including an exhaustive analysis of an explicit example of inverse problem; and Section 4 shows other applications of the techniques developed in the paper, i.e., a complete classification of the pairs of orthogonal polynomials related by certain type of linear relations with constant polynomial coefficients, and the determination of the orthogonal polynomials whose associated ones come from a polynomial modification of degree one of the original orthogonality functional.

Now we proceed with the conventions for the notation.

In what follows ${\mathbb{T}}:=\{z\in{\mathbb{C}}:|z|=1\}$ and ${\mathbb{D}}:=\{z\in{\mathbb{C}}:|z|<1\}$ are called respectively the unit circle and the open unit disk on the complex plane. ${\mathbb{P}}:={\mathbb{C}}[z]$ is the complex vector space of polynomials with complex coefficients, and ${\mathbb{P}}_{n}$ the vector subspace of polynomials whose degree is not greater than $n$, while ${\mathbb{P}}_{-1}:=\{0\}$ is the trivial subspace. $\Lambda:={\mathbb{C}}[z,z^{-1}]$ is the complex vector space of Laurent polynomials and, for $m\leq n$, we define the vector subspace $\Lambda_{m,n}:=\mathrm{span}\{z^{m},z^{m+1},\dots,z^{n}\}$. Given any $f\in\Lambda$ we define $f_{*}(z)=\overline{f}(z^{-1})$ and, if $p\in{\mathbb{P}}_{n}\setminus{\mathbb{P}}_{n-1}$, $p^{*}$ denotes its reversed polynomial $p^{*}(z)=z^{n}p_{*}(z)$. Sometimes we use the notation $p^{*}(z)=z^{n}p_{*}(z)$ for polynomials $p\in{\mathbb{P}}_{n}$ whose degree can be smaller than $n$. Then we refer to the $*_{n}$ operator when it is advisable to avoid misunderstandings.

Any hermitian linear functional $v$ on $\Lambda$ ($v[z^{-n}]=\overline{v[z^{n}]}$, $n=0,1,\dots$) defines a sesquilinear functional $(\cdot\;,\cdot)_{v}\colon\Lambda\times\Lambda\longrightarrow{\mathbb{C}}$ by

[TABLE]

The sequence $(p_{n})_{n\geq 0}$ is a sequence of orthogonal polynomials with respect to the hermitian linear functional $v$ if

• (i)

$p_{n}\in{\mathbb{P}}_{n}\setminus{\mathbb{P}}_{n-1}$,

• (ii)

$(p_{n},p_{m})_{v}=l_{n}\delta_{n,m},\quad l_{n}\neq 0$,

and when such a sequence exists $v$ is called a quasi-definite functional. If $v[1]\neq 0$ we can assure only the existence of a finite segment of orthogonal polynomials, i.e., a finite set $(p_{k})_{k=0}^{n}$ of polynomials satisfying (i) and (ii). When $v$ has a finite segment of orthogonal polynomials $(p_{k})_{k=0}^{n}$ of length $n+1$ we say that $v$ is quasi-definite on ${\mathbb{P}}_{n}$.

In the positive definite case ($l_{n}>0$, $n=0,1,\dots$) there exists a positive measure $\mu$ supported on ${\mathbb{T}}$ providing an integral representation for the functional $v$,

[TABLE]

Due to this reason a sequence $(p_{n})_{n\geq 0}$ satisfying (i) and (ii) is called a sequence of orthogonal polynomials on the unit circle, even in the general quasi-definite case. If $l_{n}=\pm 1$ for all $n$, $(p_{n})$ is called a sequence of orthonormal polynomials on the unit circle. We denote by $(\hat{p}_{n})_{n\geq 0}$ the orthonormal polynomials with positive leading coefficients.

In that follows $(\psi_{n})_{n\geq 0}$ denotes the sequence of monic orthogonal polynomials (MOP) with respect to a hermitian functional $v$. Two hermitian linear functionals $v_{1},v_{2}$ have a common finite segment $(\psi_{j})_{j=0}^{n}$ of MOP iff there exists $\lambda\in{\mathbb{R}}^{*}$ such that $v_{1}[f]=\lambda v_{2}[f]$ for any $f\in\Lambda_{-n,n}$, although requiring this condition to hold only for any $f\in{\mathbb{P}}_{n}$ is enough due to the hermiticity. In this case we say that $v_{1}$ and $v_{2}$ are equivalent in ${\mathbb{P}}_{n}$ or, in a more symbolic way, $v_{1}\equiv v_{2}$ in ${\mathbb{P}}_{n}$. If this holds for any $n$, we simply say that $v_{1}$ and $v_{2}$ are equivalent and we write $v_{1}\equiv v_{2}$.

A sequence $(\psi_{n})$ is a sequence of MOP on the unit circle iff it satisfies the recurrence relation (see [25, 13, 22])

[TABLE]

with $\psi_{0}(z)=1$ and $|\psi_{n}(0)|\neq 1$ for $n\geq 1$. Applying the $*_{n}$ operator to the above recurrence we get the equivalent one

[TABLE]

The values $\psi_{n}(0)$ are called the Schur parameters or reflection coefficients of the hermitian linear functional $v$.

A straightforward computation yields

[TABLE]

where $\varepsilon_{n}:=(\psi_{n},\psi_{n})_{v}=v[\psi_{n}z^{-n}]$ relates $\hat{p}_{n}$ and $\psi_{n}$ by $\hat{p}_{n}=|\varepsilon_{n}|^{-{1\over 2}}\psi_{n}.$ When $v$ is positive definite $\varepsilon_{n}=\|\psi_{n}\|^{2}_{L^{2}(\mu)}>0$ for $n\geq 0$, which means that $|\psi_{n}(0)|<1$ for $n\geq 1$.

2 Hermitian polynomial modifications

We are interested in those (Laurent) polynomial modifications of hermitian functionals which preserve their hermitian character, in short, the hermitian polynomial modifications of hermitian functionals. If $v$ is a linear functional on $\Lambda$ and $L\in\Lambda$ the modified functional $vL$ is defined by

[TABLE]

The modified functional $vL$ is hermitian for every hermitian $v$ iff $L_{*}=L$, which is equivalent to state that $L=P+P_{*}$ with $P\in{\mathbb{P}}$ (see [2]). Such a polynomial $P$ can be uniquely determined by $L$ simply requiring $P(0)\in{\mathbb{R}}$, a convention that we will assume in what follows. We will refer to $\deg P$ as the degree of the polynomial modification, which we will consider greater than or equal to one, and $L$ will be called a hermitian Laurent polynomial of degree $r$.

Another way to characterize a hermitian polynomial modification is through the polynomial $A=z^{\deg P}L$ of degree $2\deg P$. The condition $L_{*}=L$ means that $A$ is self reciprocal, i.e., $A^{*}=A$. Thus the hermitian polynomial modifications are related to the self-reciprocal polynomials of even degree.

The set of roots of a self-reciprocal polynomial, counting the multiplicity, is invariant under the transformation $\zeta\to 1/\overline{\zeta}$. That is, their roots lie on the unit circle or appear in symmetric pairs $\zeta,1/\overline{\zeta}$. Indeed, this property characterizes the self-reciprocal polynomials up to numerical factors. This implies that any self-reciprocal polynomial of even degree factorizes into a product of self-reciprocal polynomials of degree 2. As a consequence, an arbitrary hermitian polynomial modification is a composition of elementary ones of degree 1, i.e., if $L=P+P_{*}$ with $\deg P=r$, then $L=L_{1}L_{2}\cdots L_{r}$ with $L_{k}=P_{k}+P_{k*}$ and $\deg P_{k}=1$.

Sometimes we will deal with polynomials $A\in{\mathbb{P}}_{n}$ whose degree is not necessarily $n$ but such that $A^{*_{n}}=A$. In this case we will say that $A$ is self-reciprocal in ${\mathbb{P}}_{n}$ to avoid misunderstandings. Such a polynomial has the general form $A(z)=z^{s}B(z)$ where $B$ is strictly self-reciprocal. Thus, a self-reciprocal polynomial in ${\mathbb{P}}_{n}$ is actually self-reciprocal iff it has no zeros at the origin.

Given a hermitian functional $v$ and a Laurent polynomial $L=P+P_{*}$, our purpose is to obtain relations between the MOP and Schur parameters associated with the functionals $v$ and $vL$. Multiplying $L$ by a non null real factor gives rise to a hermitian functional which is equivalent to $vL$ and, hence, with the same MOP and Schur parameters as $vL$. Therefore, concerning our aim, the Laurent polynomial $L$, as well as the polynomials $P$ and $A$, are defined up to non null real factors.

The following general result will be useful to achieve our objective. In what follows we denote by $S^{\bot_{n}}$ the orthogonal complement in ${\mathbb{P}}_{n}$ of a subspace $S\subset{\mathbb{P}}_{n}$.

Lemma 2.1** (see [26]).**

Let $v$ be a hermitian functional such that the corresponding $n$-th MOP $\psi_{n}$ exists. Then, $\mathfrak{B}=\{z^{k}\psi_{n}\}_{k=0}^{r}\cup\{z^{k}\psi_{n}^{*}\}_{k=0}^{r-1}$ is a basis of \bigl{(}z^{r}\mathbb{P}_{n-r-1}\bigr{)}^{\bot_{n+r}} for $n\geq r\geq 1$, and a generator system of ${\mathbb{P}}_{n+r}$ for $r>n\geq 0$.

Sketch of the proof. If $n\geq r\geq 1$, the orthogonality of $\psi_{n}$ assures that \mathfrak{B}\subset\bigl{(}z^{r}\mathbb{P}_{n-r-1}\bigr{)}^{\bot_{n+r}}. Besides, $\Omega\in\mathrm{span}\mathfrak{B}$ iff $\Omega=C\psi_{n}+D\psi_{n}^{*}$, $C\in{\mathbb{P}}_{r}$, $D\in{\mathbb{P}}_{r-1}$. Furthermore, this decomposition is unique because $\gcd(\psi_{n},\psi_{n}^{*})=1,$ which proves the linear independence of $\mathfrak{B}.$ Then, the first result follows from the fact that \sharp\mathfrak{B}=2r+1=\dim\bigl{(}z^{r}\mathbb{P}_{n-r-1}\bigr{)}^{\bot_{n+r}}.

Suppose now that $r>n\geq 0$. From the previous result we know that $\{z^{k}\psi_{n}\}_{k=0}^{n}\cup\{z^{k}\psi_{n}^{*}\}_{k=0}^{n-1}$ is a basis of $\mathbb{P}_{2n}$. Hence, $\{z^{k}\psi_{n}\}_{k=0}^{r}\cup\{z^{k}\psi_{n}^{*}\}_{k=0}^{n-1}$ is a linear independent subset of ${\mathbb{P}}_{n+r}$ with $n+r+1$ elements, thus it is a basis of ${\mathbb{P}}_{n+r}$, which proves the second result. $\Box$

Our interest in the previous lemma is the following direct consequence.

Corollary 2.2**.**

Let $v$ be a hermitian functional such that the corresponding $n$-th MOP $\psi_{n}$ exists. Then, every polynomial \Omega\in\bigl{(}z^{r}\mathbb{P}_{n-r-1}\bigr{)}^{\bot_{n+r}} has a unique decomposition $\Omega=C\psi_{n}+D\psi_{n}^{*}$, $C\in{\mathbb{P}}_{r}$, $D\in{\mathbb{P}}_{r-1}$, for $n\geq r\geq 1$, and every polynomial $\Omega\in{\mathbb{P}}_{n+r}$ has infinitely many such decompositions for $r>n\geq 0$.

Remark 2.3*.*

It is worth it to remark the case $n=r$ in the above corollary, which says that every polynomial $\Omega\in\mathbb{P}_{2r}$ admits a unique decomposition $\Omega=C\psi_{r}+D\psi_{r}^{*}$, $C\in{\mathbb{P}}_{r}$, $D\in{\mathbb{P}}_{r-1}$.

The next theorem is the starting point for our approach to the study of hermitian polynomial modifications of hermitian functionals.

Theorem 2.4**.**

Let $u$, $v$ be hermitian functionals with finite segments of MOP $(\varphi_{j})_{j=0}^{n}$, $(\psi_{j})_{j=0}^{n+r}$ respectively, and let $L=P+P_{*}=z^{-r}A$ with $P$ a polynomial of degree $r$. Then, the following statements are equivalent:

• (i)

$u\equiv vL$* in ${\mathbb{P}}_{n}$.*

• (ii)

There exist $C_{j}\in{\mathbb{P}}_{r}$, $D_{j}\in{\mathbb{P}}_{r-1}$ with $C_{j}(0)\neq 0$ such that

[TABLE]

• (iii)

There exist $C_{j}\in{\mathbb{P}}_{r}$, $D_{j}\in{\mathbb{P}}_{r-1}$ with $C_{j}(0)\neq 0$ such that

[TABLE]

The polynomials $C_{j}\in{\mathbb{P}}_{r}$, $D_{j}\in{\mathbb{P}}_{r-1}$ satisfying $(\ref{GPOLO})$ or $(\ref{GPOLO1})$ are unique, $\deg C_{j}=r$, $C_{j}(0)\in{\mathbb{R}}$ and $C_{j}^{*}(0)=A(0)$.

Proof.

The equivalence between (ii) and (iii) follows from the use of the $*_{2r+j}$ operator and the fact that $A$ is a self-reciprocal polynomial of degree $2r$. Also, assuming (ii) we get $\deg C_{j}=r$ because $\deg(D_{j}\psi_{j+r}^{*})<\deg(A\varphi_{j})=2r+j$, and the equality (\varphi_{j},\varphi_{j})_{u}=u\bigl{[}\varphi_{j}z^{-j}\bigl{]}=C_{j}(0)\varepsilon_{j+r} implies $C_{j}(0)\in{\mathbb{R}}$. On the other hand, evaluating (4) at $z=0$ we find that $C_{j}^{*}(0)=A(0)$. It only remains to prove the equivalence between (i) and (ii) and the uniqueness of decomposition (3).

Suppose (i), i.e., $u[f]=\lambda vL[f]$, $\lambda\in{\mathbb{R}}^{*}$, for any $f\in\Lambda_{-n,n}$. The orthogonality of $(\varphi_{j})_{j=0}^{n}$ with respect to $u$ gives

[TABLE]

which means that A\varphi_{j}\in\bigl{(}z^{r}\mathbb{P}_{j-1}\bigr{)}^{\bot_{2r+j}} with respect to $v$. Using Corollary 2.2 we get (3) and the uniqueness of the polynomials $C_{j}$, $D_{j}$.

On the other hand, if $(\varphi_{j})_{j=0}^{n}$, $(\psi_{j})_{j=0}^{n+r}$ satisfy (3), the orthogonality of $(\psi_{j})_{j=0}^{n+r}$ with respect to $v$ yields

[TABLE]

for $0\leq k\leq j-1$ and

[TABLE]

So, $C_{j}(0)\neq 0$ for $j=0,\dots,n$ iff $(\varphi_{j})_{j=0}^{n}$ is a finite segment of MOP with respect to $vL$, which means that $u\equiv vL$ in ${\mathbb{P}}_{n}$. ∎

Equality (4) is true taking $D_{j}^{*}=D_{j}^{*_{r-1}}$, no matter whether $D_{j}$ has degree $r-1$ or not. In what follows we will assume this convention for the polynomials $D_{j}$.

Remark 2.5*.*

The functional $u$ has a finite segment of MOP of length (at least) one iff $u[1]\neq 0$. Therefore, Theorem 2.4 assures that the condition $v[L]\neq 0$ is equivalent to the existence of a (unique) decomposition

[TABLE]

with $C_{0}(0)\neq 0$. However, Remark 2.3 says even more: no matter the value of $v[L]$, there is always a unique decomposition like (5). The equality $v[L]=C_{0}(0)\varepsilon_{r}$ implies that $v[L]\neq 0$ is only responsible of $C_{0}(0)\neq 0$.

The above theorem has the following consequence for quasi-definite functionals.

Corollary 2.6**.**

Let $u$, $v$ be quasi-definite functionals with sequences of MOP $(\varphi_{n})$, $(\psi_{n})$ respectively, and let $L=P+P_{*}=z^{-r}A$ with $P$ a polynomial of degree $r$. Then, $u\equiv vL$ iff there exist polynomials $C_{n}\in{\mathbb{P}}_{r}$, $D_{n}\in{\mathbb{P}}_{r-1}$ with $C_{n}(0)\not=0$ such that

[TABLE]

or equivalently

[TABLE]

For convenience, in what follows we will use a matrix notation and we will adopt some definitions and conventions that will be used in the rest of the paper. If $L$ is a hermitian Laurent polynomial of degree $r$, $P$ and $A$ are the polynomials given by $L=P+P_{*}=z^{-r}A$, $P(0)\in{\mathbb{R}}$. We denote by $\phi_{j}$ and $\psi_{j}$ the $j$-th MOP with respect to the hermitian functionals $u$ and $v$ respectively. Also,

[TABLE]

The matrices $\mathcal{S}_{j}$ and $\mathcal{T}_{j}$, known as transfer matrices, permit us to write recurrence relations (1) and (2) for $(\varphi_{n})$ and $(\psi_{n})$ in the compact form

[TABLE]

while the matrices $\mathcal{C}_{j}$ make possible to combine (3) and (4) into

[TABLE]

The structure of the matrices $\mathcal{C}_{j}$ is worth to be remarked.

Definition 2.7**.**

A polynomial matrix \mathcal{C}=\scriptsize\pmatrix{C_{1}&D_{1}\cr D_{2}&C_{2}}, $C_{i},D_{i}\in{\mathbb{P}}_{r}$, satisfying ${\mathcal{C}}^{*_{r}}=J\mathcal{C}J$ with J=\scriptsize\pmatrix{0&1\cr 1&0} will be called a $J$-self-reciprocal matrix in ${\mathbb{P}}_{r}$. This is equivalent to state that $C_{2}=C_{1}^{*_{r}}$ and $D_{2}=D_{1}^{*_{r}}$.

We denote by ${\mathbb{J}}_{r}$ the set of $J$-selfreciprocal matrices in ${\mathbb{P}}_{r}$ such that $C_{2}(0)\neq 0$ and $D_{2}(0)=0$. These conditions mean that $\deg C_{1}=r$ and $\deg D_{1}\leq r-1$, thus the general form of a polynomial matrix $\mathcal{C}\in{\mathbb{J}}_{r}$ is

[TABLE]

where here and below we assume that $D^{*}=D^{*_{r-1}}$.

Given a polynomial matrix $\mathcal{C}\in{\mathbb{J}}_{r}$ like (8) we will denote

[TABLE]

which is $J$-self-reciprocal too, but in general does not necessarily belong to ${\mathbb{J}}_{r}$ because $zD$ can have degree $r$.

The determinant of a $J$-self-reciprocal matrix $\mathcal{C}$ in ${\mathbb{P}}_{r}$ is a self-reciprocal polynomial in ${\mathbb{P}}_{2r}$. When $\det\mathcal{C}$ has degree $2r$ we will say that $\mathcal{C}$ is a regular $J$-self-reciprocal matrix. This is equivalent to $\det\mathcal{C}(0)\neq 0$, which in case of $\mathcal{C}\in{\mathbb{J}}_{r}$ means simply $C(0)\neq 0$. We will denote by ${\mathbb{J}}_{r}^{\mathrm{reg}}$ the subset of regular $J$-self-reciprocal matrices of ${\mathbb{J}}_{r}$.

The next result about $J$-self-reciprocal matrices will be useful later on.

Lemma 2.8**.**

Let \mathcal{S}=\pmatrix{z&a\cr z\overline{a}&1}, \mathcal{T}=\pmatrix{z&b\cr z\overline{b}&1} with $a,b\in{\mathbb{C}}$.

• (i)

If $|a|\neq 1$, $\mathcal{C}\in{\mathbb{J}}_{r}$, the equation $\mathcal{C}\mathcal{T}=\mathcal{S}\hat{\mathcal{C}}$ defines a matrix $\hat{\mathcal{C}}\in{\mathbb{J}}_{r}$ iff

[TABLE]

In this case $\hat{\mathcal{C}}\in{\mathbb{J}}_{r}^{\mathrm{reg}}\;\Leftrightarrow\;|b|\neq 1,\;\mathcal{C}\in{\mathbb{J}}_{r}^{\mathrm{reg}}$.

• (ii)

If $|b|\neq 1$, $\mathcal{C}\in{\mathbb{J}}_{r}$, the equation $\hat{\mathcal{C}}\mathcal{T}=\mathcal{S}\mathcal{C}$ defines a matrix $\hat{\mathcal{C}}\in{\mathbb{J}}_{r}$ iff

[TABLE]

In this case $\hat{\mathcal{C}}\in{\mathbb{J}}_{r}^{\mathrm{reg}}\;\Leftrightarrow\;|a|\neq 1,\;\mathcal{C}\in{\mathbb{J}}_{r}^{\mathrm{reg}}$.

Proof.

If $|a|\neq 1$ the equation $\mathcal{C}\mathcal{T}=\mathcal{S}\hat{\mathcal{C}}$ can be written as

[TABLE]

Let $\mathcal{C}\in{\mathbb{J}}_{r}$. Then $\mathcal{X}$ is a $J$-self-reciprocal matrix in ${\mathbb{P}}_{r}$