On Linear Combinations of Two Orthogonal Polynomial Sequences on the Unit Circle
© C. Suárez. 2010
Received: 1 August 2009
Accepted: 5 March 2010
Published: 8 March 2010
Let be a monic orthogonal polynomial sequence on the unit circle. We define recursively a new sequence of polynomials by the following linear combination: , , . In this paper, we give necessary and sufficient conditions in order to make be an orthogonal polynomial sequence too. Moreover, we obtain an explicit representation for the Verblunsky coefficients and in terms of and . Finally, we show the relation between their corresponding Carathéodory functions and their associated linear functionals.
1. Notation and Preliminary Results
Along this paper, we will use the following notations. We denote by the linear space of Laurent polynomials with complex coefficients and by the dual algebraic space of . Let be the space of complex polynomials.
For simplicity, along this paper we also assume that is normalized (i.e., ). It is well known that the regularity of is a necessary and sufficient condition for the existence of a sequence of orthogonal polynomials on the unit circle. On the other hand, the polynomials satisfy the so-called Szegö recurrence relations
This sequence verifies the following properties:
The measure can be reconstructed from by means of the inversion formula. The aim of this paper is the analysis of the following problem. Given an MOPS on the unit circle , orthogonal with respect to a linear functional , to find necessary and sufficient conditions in order to make a sequence of monic polynomials defined by
For instance, this subject has been treated in [4–6] in the context of the theory of orthogonal polynomials on the real line. For orthogonal polynomials with respect to measures supported on the unit circle, in  there have been relevant results.
The structure of this paper is the following. In Section 2 we give the necessary conditions in order to be sure that the problem (1.15) admits a nontrivial solution. In Section 3, we prove a sufficient condition and we obtain the explicit solution in terms of and . Section 4 is devoted to find the functional relation between and . Finally, Section 5 contains the rational relation between the corresponding Carathéodory functions.
2. Necessary Conditions
The following proposition justifies this choice.
Under the same conditions as in Proposition 2.1, the following assertions hold:
The result follows from (2.6) and the above item.
The next result will be used later.
This number plays a very important role in the solution of our problem.
By solving this, we get (2.17).
Let us proceed with the proof of (ii). Inserting
Using (2.28) we obtain
This complex constant is denoted in the statement by . The property is a consequence of Corollary 2.2(iii). On the other hand, the explicit expressions of and follow from (2.7) for and , respectively.
Finally, we show (iii). Using again (2.17), we have
Substituting this relation in (2.31) and using the recurrences of the kernels (1.9) and (1.10), (2.22) holds.
In order to state the converse we need the following assertions.
and it is true according to (2.42).
3. Some Solutions
It only remains to establish the sufficient condition.
We first show that (3.6) implies (3.1)
Therefore, the result follows immediately from (3.5).
and this is equal to zero from (3.3).
Now, we are going to express the Verblunsky coefficients for the solutions in terms of and . We remember that to give a MOPS on the unit circle is equivalent to know the sequence of complex numbers with .
This completes the proof because of the symmetry of the problem.
Notice that the restrictions given for and in the previous theorem ensure that the sequences generated by and are MPOS, but they do not ensure that and fulfill (3.1). In fact, other similar conditions to (2.42) seem to be necessary in order to obtain a characterization of the Verblunsky coefficients in terms of and .
4. Linear Functionals
where the last equality follows from (2.34).
We compute the right-hand side using(2.31)
The opposite question has been proved in . That is, if and are regular functionals related by (4.1) with , then the corresponding orthogonal polynomials satisfy (2.1).
5. Carathéodory's Functions
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