It is very well known that the orthogonal polynomials associated with a positive Borel measure supported on the real line (OPRL) have many properties that allow us, both to emulate the measure, and describe its support. For example, the zeros of are simple, they lie on the open convex hull of and there is one zero on the closure of each connected component of the complement of , at most. Moreover, the zeros of and interlace and in general, they end up filling the whole support. These zeros describe as because they allow us to construct a sequence of measures that converges weakly to .
Orthogonal polynomials on the unit circle (OPUC), or Szegő polynomials, were introduced in (Sz, , Chapter 11). Since then, they have been widely studied, not only because of their own interest SimonBk
, but also in many applications such as the trigonometric moment problemAkh , complex approximation Walsh
, probability and statisticsGS , prediction theory WM , systems theory, networks, circuits and scattering DD , signal processing Del , and many more, but also because of their intimate connection with OPRL (see e.g. Bul ; CDPPablo
). However, these polynomials present important differences with respect to OPRL, in particular, concerning the above properties, since their zeros are located outside of the support of the measure. As a consequence, these zeros cannot be used for interpolation processes on the unit circle, as nodes for quadrature formulas on the unit circle, and they lack importance in the resolution of the trigonometric moment problem, among other relevant properties that hold for OPRL. These drawbacks were not solved until the concept of paraorthogonality (JNT , see also (Ger, , Theorem III)) was introduced in this setting. These paraorthogonal polynomials (POPUC) have their zeros in the support of the measure when the support is . In this context, focussing on their spectral properties, quadrature, and associated problems, these polynomials are a natural counterpart on the unit circle of the orthogonal polynomials on the real line. Some of the properties that we have mentioned above and many others formulated for a measure supported on the real line, have been adapted for measures on the unit circle, see e.g. CMV02 ; G ; S ; Wong .
Since the spectra of the orthogonal and paraorthogonal polynomials are directly connected to their respective type of orthogonality, we shall also look at what can be said about the zeros as we relax some of the orthogonality conditions of these polynomials, leading to quasi-orthogonal polynomials on the real line and the quasi-paraorthogonal counterpart on the unit circle.
In the case of the real line, the real polynomials of degree that are orthogonal to
(the vector space of all polynomials of degree at most), are well studied and they are called quasi-orthogonal polynomials (QOPRL) of degree and order or -QOPRL. Clearly the orthogonal polynomial appears as a special case for : . The QOPRL have at least changes of sign on the open convex hull of . This concept was first introduced in 1923 by M. Riesz (Rie ) for in relation to moment problems, and then by L. Fejèr () and by J.A. Shohat () in 1933 and 1937, respectively (Fej ; Sho ), in the context of quadrature formulas (see also Chi2 ; Dic ; Dra ).
The unit circle analogue of these QOPRL should naturally be called quasi-paraorthogonal polynomials (QPOPUC). Their properties however, are less developed in the literature. This is one of the main purposes of this paper: to define and analyse the spectral properties of these QPOPUC and how these relate to their orthogonality conditions.
The outline of the paper is as follows. In Section 2 we introduce the concept of quasi-paraorthogonal polynomials on the unit circle, where we shall generalize the classical concept of paraorthogonality to QPOPUC of order by requiring orthogonality to spaces of decreasing dimension. The classical concept of paraorthogonality corresponds to quasi-paraorthgonality of order 1. We prove the main result in Section 3
: a link between the number of paraorthogonality conditions and the number of zeros of odd multiplicity in the interior of the support of the measure, when it is supported on an arc of the unit circle. We also investigate how we can use the free parameters to prefix some of the zeros. As an application we consider in Section4 how this can be used to construct positive Szegő-type quadrature formulas on an arc or on that are the analogues of the Gauss-type formulas on an interval or . Finally, in Section 5 we shall include some numerical experiments to illustrate these quadrature formulas.
We finish this introductory section by defining some notation and abbreviations that are used throughout this paper. We denote by the vector space of all polynomials and by the -dimensional space of polynomials of degree at most . The vector space of all Laurent polynomials is , with subspaces , , . is the real line and the complex unit circle and is a positive Borel measure that is supported on a subset of or . If or we assume it is compact and connected, i.e. an interval on or an arc on . We denote by the inner product induced by and we assume that belongs to the corresponding Hilbert space . Orthogonality will always refer to this space. The acronyms OPRL and OPUC are used for orthogonal polynomials on the real line and the unit circle respectively. POPUC refers to paraorthogonal polynomials on the unit circle and if the letter Q is added in the beginning, it refers to quasi-versions, thus QOPRL means quasi-orthogonal polynomials on the real line and QPOPUC denotes quasi-paraorthogonal polynomials on the unit circle. The precise definitions follow below. By and we denote the interior and exterior of the closed unit disk, respectively and by the integer part of . is the reciprocal (or reversed) polynomial of . Throughout the paper, will denote the monic orthogonal polynomial of degree in . For , the Schur parameters for the OPUC will be denoted by where and , i.e., for . All our spaces will be subspaces of , and when we write , we mean its orthogonal complement in . We denote by , the Möbius Transform associated with . We also assume that is 0 and is 1 if .
2 Quasi-paraorthogonal polynomials
As we have already said in the Introduction, quasi-orthogonal polynomials with respect to a measure on the real line have real coefficients, and as a consequence, their zeros are real or appear in complex conjugate pairs (symmetric with respect to ). The analogue on the unit circle, is that zeros are on or appear in symmetric pairs with respect to . That is: if is a zero then also is a zero. Note that this implies that if , then cannot be a zero of . Thus, if the zeros are , , then we are looking for polynomials satisfying
The notation is somewhat standard in the OPUC literature to denote the reciprocal of a polynomial . We have indeed that if then . This explains the following definition.
Definition 2.1 (invariant).
We say that the polynomial is invariant if and only if it satisfies , . The parameter is called the invariance parameter and is said to be -invariant.
Note that if is a monic invariant polynomial, then its invariance parameter equals and if , and is -invariant, then is -invariant with .
Thus, since we are interested in the symmetry of the zeros, we should only consider invariant polynomials. Szegő polynomials have all their zeros in and thus they are not invariant. Quasi versions are obtained by imposing certain orthogonality conditions to subspaces of dimension . What should this subspace like for an invariant polynomial ?
Let be an invariant polynomial on . Then
Thus , for . An invariant polynomial on should always be orthogonal to a subspace that is spanned by powers of that are centrosymmetric in the sequence , that is, if it includes , it should include as well. Note that for QOPRL the number of orthogonality conditions is reduced by 1 as the order increases by 1. However the symmetry in paraorthogonality implies that if we remove orthogonality to then we also remove orthogonality to . The number of orthogonality conditions, and thus also the order of QPOPUC, changes in steps of two. So we define a nested sequence of subspaces of dimension in :
Invariant polynomials, orthogonal to this subspace for were coined paraorthogonal as they were introduced in JNT , (see also (Ger, , Theorem III)), and they were later studied by many DG91a ; DG91b ; CMV02 ; CMV06 ; CMNR16 ; CCBPP16 ; G ; Wong ; MISCFSS19 ; NS . We shall refer to them as quasi-paraorthogonal polynomials (QPOPUC) of order 1. More generally, we define the set of -QPOPUC of order as all invariant polynomials of degree in . The prefix quasi refers to the fact that they satisfy less paraorthogonality (i.e., symmetric orthogonality) conditions.
We recover a representation of -QPOPUC in terms of the orthogonal polynomials obtained by Peherstorfer in Peh .
The monic -QPOPUC are given by
The set of all monic -QPOPUC depends on real free parameters.
We first note that has dimension . Furthermore, , for , . Thus the polynomials are in . Moreover they are independent because is only possible if . Indeed, if , then , which is impossible because the rational function is irreducible of degree since and have no common zeros and the rational function has degree at most . We get a similar contradiction if so that .
Let be in with monic, then is monic of degree . Because the representation w.r.t. a basis is unique, is -invariant if and only if . Therefore . The parameters represented by that depends on a real parameter and by the complex coefficients , that correspond to real parameters. ∎
The monic polynomial can be written as
So we shall in the rest of the paper often use the zeros as parameters instead of the free coefficients of the monic polynomial .
If , then (see Lemma A.3 of the Appendix) but there will be some ‘extra’ orthogonality. Because with . Define , then clearly . Thus every -QPOPUC with is orthogonal to . We shall refer to as the orthogonality parameter of .
If then the orthogonality parameter can be made explicit in an alternative expression for as explained in Lemma A.4 of the Appendix. If , then and thus is always nonzero.
3 Zeros of quasi-paraorthogonal polynomials
In the previous section we have described all monic and invariant quasi-paraorthogonal polynomials. We are now in a position to prove the main result that is an analogue on the unit circle of a very well known important result when dealing with measures supported on the real line that connects the number of orthogonality conditions with the number of zeros of odd multiplicity in the interior of the support of the measure.
It has been shown in the literature (see Sho ; Jou ; Mon ; BCBVB10 ; BBMCQ13 ) that an -QOPRL associated with a measure whose support is an interval has at least zeros of odd multiplicity in . For we can use the parameter to fix one zero in , for , there are two parameters that can be used to fix and as nodes and for one may fix , and some , while in all these cases it is guaranteed that there are simple zeros in the support of the measure. These QOPRL for or are mainly derived in the context of Gauss-Radau and Gauss-Lobatto quadrature formulas respectively.
In this section we shall obtain similar results for QPOPUC assuming that the measure is supported on an arc .
3.1 A general statement
The next Lemma gives a property of invariant polynomials that will be of interest for our purposes.
Let be an invariant polynomial. Then,
Factoring structure on : if we denote by the zeros of of odd multiplicity on , then
where and are zeros of that may possibly coincide with .
If is odd, then the number of zeros of odd multiplicity on is odd. If is even, then the number of zeros of odd multiplicity on is even (possibly zero).
The second property is a direct consequence of the first one. If is a zero of , then is also a zero of . So, the factorization of will have the factors
If is a zero of on and it has multiplicity at least two, then will have among its factors
We shall denote an oriented arc as an interval running counterclockwise from to . If denotes a (closed) arc, then the complementary arc with the same orientation is . The square or rounded brackets are used as in the case of a real interval to indicate that boundary points are included or not. Moreover, an arc can be indicated by three points like arc, then this defines its orientation: from over to , and we write .
For , , and , an -QPOPUC has at least zeros of odd multiplicity on the unit circle and of them in .
Let us first consider the case . Since a QPOPUC is invariant, we know from Lemma 3.1 that has an odd number of zeros of odd multiplicity on and that it can be factored as
with odd. We have distinguished between the with even multiplicity and for those of odd multiplicity we have the zeros that are in and the distinct that are not in , that is . Without loss of generality we order the such that . We define the arcs with midpoints and for .
Our aim is to prove that which we do by contraposition, thus suppose that . We first treat the case where is even. Thus suppose , , then
By Lemma A.1,
so that , has a constant sign in .
But because . Indeed, since , it only remains to prove that and that . The assumptions are , , and (notice that is odd). So,
This proves that has at least zeros of odd multiplicity in , and since is odd, has zeros on the unit circle.
Now, we consider , then is even and . We consider
By the same argument, has a constant sign in , which is a contradiction since and thus . Indeed, and now, ( is even) and we have supposed that , hence . So, and . Again in this case . This proves that has at least zeros of odd multiplicity in , and since is odd, has zeros on the unit circle.
When is even the proof is similar. ∎
3.2 and one prescribed zero
If we can employ the invariance parameter (i.e. for a monic ) to place an extra zero of odd multiplicity on , then this would imply that the simple zeros of are in . We shall study the situation using Blaschke products. Since the zeros of the Szegő polynomial lie in , we can define the Blaschke products of degree (with zeros in ) as follows
By the Argument Principle, goes around the unit circle exactly times when wraps around the origin (a -to-1 map), and since all their zeros are in the map preserves the orientation. If in addition the support of the measure is an arc , as it is in our case, the following result is a consequence of the previous theorem.
Let be a positive measure supported on an arc with associated orthogonal polynomials and let be the Blaschke product as defined above in (4). Then for given , will have at least solutions in .
Let be a positive measure supported on , and . Consider the QPOPUC and define and with as in (4). Then has all its simple zeros in (or in ) if and only if (or respectively). Moreover is one of these zeros if and only if .
The zeros of are the roots of the equation . We denote the (ordered) roots of by and the (ordered) roots of , by , with and . By the properties of , , . ∎
Note also that the only prefixed zero that is allowed must be in one of the intervals , . Compare with (BCBVB10, , Theorem 2.9) for QOPRL.
3.3 and 2 or 3 prescribed zeros
A monic QPOPUC
of order 3, has at least zeros of odd multiplicity on , of them are in interior of the arc where the measure is supported, and it depends on three real parameters that can be used to fix up to three zeros, ensuring that all the zeros of are simple and placed on the support of measure. Let us start by fixing 2 nodes.
If , then is a Blaschke product of degree and has simple solutions on . The conditions lead to
This gives the system
If , this represents two secants for the circle
The first secant passes through and and the second passes through and . If , then , and this should be excluded because must be in . These secants intersect in exactly one finite point if and only if and the intersection point is
By Lemma A.2 of the Appendix, if and only if the arcs and have opposite orientation. (See also Figure 1.) Note that we can only define and as arcs if and which is implicitly assumed also in the next Theorem.
Let . Given , and distinct . Define as in (7). Assume the are distinct and . Then the monic -QPOPUC of (5) is uniquely defined and has all its zeros simple and on , and being two of them if is such that the arcs and have opposite orientation. The parameter in (5) belongs to and depends on as given by (10).
Note that there are infinitely many that allow us to fix and ensuring that has all its zeros simple on .
The from (10) corresponds to where . For we get and for we get . Thus . Since there are two different points of intersection, this means that there are values for that will deliver and others for which . It is clear that the line connecting the origin with , i.e., the mediatrix of the chord , will intersect at the points and the circle at two points: one in that is closest to the origin and one in that is the farthest from the origin. The former is given by , which is with . Thus if and only if is in the interior of the for which describes the . Taking into account the value of , if and only if .