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An extension of the associated rational functions on the unit circle

A special class of orthogonal rational functions (ORFs) is presented in this paper. Starting with a sequence of ORFs and the corresponding rational functions of the second kind, we define a new sequence as a linear combination of the previous ones, the coefficients of this linear combination being self-reciprocal rational functions. We show that, under very general conditions on the self-reciprocal coefficients, this new sequence satisfies orthogonality conditions as well as a recurrence relation. Further, we identify the Caratheodory function of the corresponding orthogonality measure in terms of such self-reciprocal coefficients. The new class under study includes the associated rational functions as a particular case. As a consequence of the previous general analysis, we obtain explicit representations for the associated rational functions of arbitrary order, as well as for the related Caratheodory function. Such representations are used to find new properties of the associated rational functions.


Introduction
Since the fundamental work of Stieltjes and Chebyshev, among others, in the 19 th century, orthogonal polynomials (OPs) have been an essential tool in the analysis of basic problems in mathematics and engineering. For example, moment problems, numerical quadrature, rational and polynomial approximation and interpolation, linear algebra, and all the direct or indirect applications of these techniques in engineering are all indebted to the basic properties of OPs. Mostly orthogonality has been considered on the complex unit circle or on (a subset of) the real line.
Orthogonal rational functions (ORFs) were first introduced by Džrbašian in the 1960s. Most of his papers appeared in Russian literature, but an accessible survey in English can be found in [18,21]. These ORFs are a generalization of OPs in such a way that they are of increasing degree with a given sequence of poles, and the OPs result if all the poles are at infinity. During the last years, many classical results of OPs are extended to the case of ORFs.
Several generalizations for ORFs on the complex unit circle and the whole real line have been gathered in book [4, Chapt. 2-10] (e.g. the recurrence relation and the Favard theorem, the Christoffel-Darboux relation, properties of the zeros, etc.). Other rational generalizations can be found in e.g. [9,35]. Further, we refer to [2,3,7] and to [34] for the use of these ORFs in respectively numerical quadrature and system identification, while several results about matrix-valued ORFs can be found in e.g. [19,20].
Of course, many of the classical OPs are not defined with respect to a measure on the whole unit circle or the whole real line. Several theoretical results for ORFs on a subset of the real line can be found in e.g. [4,Chapt. 11] and [11,12]. For the special case in which this subset is a real half-line or an interval, we refer to [5,6] and [13,14,17,28,29,30,31] respectively, while some computational aspects have been dealt with in e.g. [15,16,25,26,27,32,33].
By shifting the recurrence coefficients in the recurrence relation for OPs and ORFs, the so-called associated polynomials (APs) and associated rational functions (ARFs) respectively are obtained. ARFs on a subset of the real line have been studied in [8,10] as a rational generalization of APs (see e.g. [24]), while APs on the complex unit circle, on the other hand, have been studied in [23]. However, so far nothing is known about ARFs on the complex unit circle, and hence, the main purpose of this paper is to generalize [23] to the case of rational functions, following the ideas developed by Professor Franz Peherstorfer.
The outline of this paper is as follows. After giving the necessary theoretical background in Section 2, in Section 3 we recall some basic properties of ORFs on the complex unit circle and their so-called functions of the second kind. Although these properties are basic, they are partially new in the sense that we prove them in a more general context. Next, in Section 4 we use these ORFs and their functions of the second kind to define a new class of ORFs on the complex unit circle. The ARFs on the complex unit circle will then turn out to be a special case of this new class of ORFs, and will be dealt with in Section 5. We conclude in Section 6 with an example.

Preliminaries
The field of complex numbers will be denoted by C, and for the real line we use the symbol R. Let a ∈ C, then ℜ{a} refers to the real part of a, while ℑ{a} refers to imaginary part. Further, we denote the imaginary unit by i. The unit circle and the open unit disc are denoted respectively by Whenever the value zero is omitted in the set X ⊆ C, this will be represented by X 0 ; e.g., For any complex function f , we define the involution operation or substar conjugate by f * (z) = f (1/z). With P n we denote the space of polynomials of degree less than or equal to n, while P represents the space of all polynomials. Further, the set of complex functions holomorphic on X ⊆ C is denoted by H(X).
Let there be fixed a sequence of complex numbers B = {β 0 , β 1 , β 2 , . . .} ⊂ D, the rational functions we then deal with, are of the form The first element β 0 has no influence in the rational functions, but it will play a role in the corresponding recurrence. The standard choice is β 0 = 0, but in this paper β 0 will be free. The reason is that, even if we choose β 0 = 0 for the orthogonal rational functions, the corresponding associated rational functions involve a shift in the poles so that the related sequence {β N , β N +1 , · · · } starts at some β N which is not necessarily zero.
Note that, whenever 1/β k = ∞ for every k 1, the "rational functions" f k (z) in (1) are in fact polynomials of degree k. Thus the polynomial case is automatically accounted for.
Note that L n and L are rational generalizations of P n and P. Indeed, if β k = 0 (or equivalently, 1/β k = ∞) for every k 0, the expression in (2) becomes ζ k (z) = z and the expression in (3) becomes B k (z) = z k . With the definition of the substar conjugate we introduce L n * = {f * : f ∈ L n }.
The superstar transformation of a complex function f n ∈ L n \ L n−1 is defined as f * n (z) = B n (z)f n * (z). Note that the factor B n (z) merely replaces the polynomial with zeros {β j } n j=1 in the denominator of f n * (z) by a polynomial with zeros 1/β j n j=1 so that L * n := {B n f * : f ∈ L n } = L n . Like in this identity, sometimes we will denote f * := B n f * when we only know that f ∈ L n , even if f could belong to L k for 1 The factors and products are named after Wilhelm Blaschke, who introduced these for the first time in [1]. some k < n. At any time, the meaning of the superstar transformation should be clear from the context. A complex function F is called a Carathéodory function (abbreviated Cfunction) in D iff and Important related functions are the Riesz-Herglotz kernel , and the Poisson kernel where the substar conjugate is with respect to t. Note that P * (t, z) = P (t, z) and P (t, z) = ℜ {D(t, z)} for z ∈ T.
To the C-function F we then associate a hermitian L F (t −k ) = L F (t k ) linear functional L F on the set of formal power series ∞ k=−∞ c k t k with complex coefficients, so that where we understand again that L F acts on t. In the remainder we will assume that F (β 0 ) = 1, and that the functional L F is positive definite. Thus, L F {1} = 1, and for every f ∈ L This is equivalent to saying that for a positive Borel measure dµ on the unit circle with T dµ(t) = 1.
We say that two rational functions f, g ∈ L are orthogonal with respect to The orthogonality φ n ⊥ F L n−1 for a function φ n ∈ L n \ {0} ensures that, in fact, φ n ∈ L n \ L n−1 .
A sequence of functions f n ∈ L n is called para-orthogonal when f n ⊥ F L n−1 (β n ) = {g ∈ L n−1 : g(β n ) = 0}. Further, a function f n ∈ L n is called k-invariant (or, self-reciprocal) iff f * n = kf n , k ∈ C. Let Φ n,τ be given by Then, it is easily verified that a self-reciprocal rational function is paraorthogonal exactly when it is proportional to a function with the form (5). Furthermore, the following theorem has been proved in [4, Thm. 5.2.1].
Theorem 1 The zeros of Φ n,τ , given by (5), are on T and they are simple.

Orthogonal rational functions and functions of the second kind
With the ORFs φ n and para-orthogonal rational functions (para-ORFs) Φ n,τ we associate the so-called functions of the second kind: (where we understand that L F acts on t) and Ψ n,τ = ψ n − τ ψ * n , τ ∈ T, respectively. We now have the following two lemmas. The first one, which is partially stated in [4, Lem. 4.2.1], can be understood as a direct consequence of the recurrence relation appearing below. The second lemma has been proved in [4,Lem. 4.2.2] for n > 0 (the statement is obvious for n = 0). 2

Lemma 2
The functions ψ n are in L n \ L n−1 .
Lemma 3 For n > 0, it holds for every f ∈ L (n−1) * and g ∈ ζ n * L (n−1) * that The same holds true for n = 0, when f, g ∈ C.
As in the polynomial case, a recurrence relation and a Favard-type theorem can be derived for ORFs and their functions of the second kind.
Theorem 4 The following two statements are equivalent: (1) φ n ∈ L n \{0} and φ n ⊥ F L n−1 , for a certain C-function F with F (β 0 ) = 1, and ψ n is the rational function of the second kind of φ n . (2) φ n and ψ n satisfy a recurrence relation of the form where λ n ∈ D, and and with initial conditions φ 0 = ψ 0 ∈ C 0 .
In the special case of orthonormality, the initial conditions are and the constants e n are given by PROOF. First, note that Thus, from the orthogonality of φ 1 , it follows that From (9) together with the definition of ψ 1 and D(t, z), we obtain where the last equality follows from (10). Further, we have that Consequently, Theorem 5 Suppose φ n ∈ L n \ {0} and φ n ⊥ F L n−1 , for a certain C-function F with F (β 0 ) = 1, and let ψ n ∈ L n \ {0} be the rational function of the second kind of φ n . Then, In the special case of orthonormality, it holds that d n = 2.
Suppose now that the equality in (11) holds true for 0 k < n with d k = 2 in the orthonormal case. We then continue by induction for k = n. From (6) it follows that so that d n ∈ R 0 and in the orthonormal case d n = d n−1 = 2, due to (8). 2 Finally, the following interpolation properties hold true for (para-)ORFs and their functions of the second kind.
Theorem 6 Suppose that F is a C-function, with F (β 0 ) = 1, and let φ n and ψ n be in L n \ {0}. Then the following two statements are equivalent: (1) φ n ⊥ F L n−1 and ψ n is the rational function of the second kind of φ n .

PROOF.
(1) ⇒ (2) has been proved in [4, Thm. 6.1.1] under the assumption β 0 = 0. The proof in [4, Thm. 6.1.1] remains valid for β 0 = 0, when replacing t and z with ζ 0 (t) and ζ 0 (z) respectively. Thus, it remains to prove that the rational functions φ n , ψ n ∈ L n \ {0} in (13) are unique up to a common nonzero multiplicative factor, as well as the fact that g n (β n ) = 0. We will prove both things simultaneously by induction on n.
From (11) it then follows that where the last equality follows from the fact that (φ * n ψ n + φ n ψ * n ) ∈ L n · L n . Consequently, which contradicts the assumption p n−1 ∈ P n−1 . 2 The following theorem directly follows from Theorem 6, and the definition of Φ n,τ and Ψ n,τ .
PROOF. The equalities in (14) have been proved in [4, Cor. 6.1.2] under the assumption β 0 = 0, but the proof remains valid for β 0 = 0. So, we only need to prove that g n (z) = 0 for every z ∈ D.
Moreover, it holds thatĝ n (β n ) = 0 for this solution. Therefore, there cannot exist rational functions R n , S n ∈ L{β 1 , . . . , β n−1 ,β n } \ {0} satisfying (15). 2 Theorem 6 is the main result of this section. It is the rational extension of [22, Thm. 2.1]. Its importance relies on the fact that it provides us with a characterization of ORFs and their second kind ones in terms only of the Cfunction F . Theorem 6 will be the key tool to study the associated ORFs and their extensions, analogously to a similar analysis of the polynomial case in [23].

A new class of orthogonal rational functions
Analogously as has been done in [23], we will study a new class of ORFs generated by a given sequence of ORFs. The rational functions of the new class will satisfy a similar recurrence to that one of the initial ORFs, but starting at some index r and with shifted poles and (rotated) parameters. The associated rational functions will be a particular case when the starting index is r = 0 and there is no rotation of the parameters. Further, we setβ 0 = β N , and hence,ζ 0 (z) = ζ N (z) andB −1 (z) = 1/ζ N (z).
This is the purpose of the following theorem.
Theorem 8 For N, n, r 0, suppose φ N +n ∈ L N +n \ {0} and φ N +n ⊥ F L N +n−1 , and let ψ N +n denote the rational function of the second kind of φ N +n . Further, suppose A, B, C and D are self-reciprocal rational functions in L N ·˜ L r , satisfying the following conditions: B(β j ) = 0, and Then the rational functions G r+n , H r+n , J r+n and K r+n , defined by are all in˜ L r+n . Furthermore, G * r+n (z) = τ A H r+n (z) and J * r+n (z) = τ A K r+n (z). (20),

PROOF. From
and .
Concerning the numerators of G r+n and H r+n , (13) and (18) give and Since the left hand side of (21) and (22) is in L N · L N ·˜ L r+n , it follows that On the other hand, taking the superstar conjugate of (22), and using the fact that A and B are self-reciprocal and satisfy (16), we obtain that , p r+n , q r+n ∈ P r+n . Since it now follows that G r+n (z) and H r+n (z) are in˜ L r+n . Further, we have that Finally, proving the statement for J r+n (z) and J * r+n (z) = τ A K r+n (z) can be done in a similar way as before, under the condition that So, it remains to prove that (23) holds true under the assumptions (16)- (19). Clearly, condition (23) holds true for N = 0. For N > 0, it follows from (18)- Condition (23) now follows due to assumption (17). 2 As a consequence of the previous theorem and Theorem 4, we have the following corollary.

Corollary 9
The rational functions G r+n and J r+n , defined as before in Theorem 8, satisfy a recurrence relation of the form where γ r+n = τ A λ N +n , v r+n (z) = c n−1 c n u N +n (z), (25) and (recall thatβ 0 = β N ) and with initial conditions G r , J r ∈˜ L r .
Theorem 8 provides us with a constructive method to generate a new class of rational functions starting with a given sequence of ORFs. As we pointed out before, the new rational functions have the same poles as the initial ORFs, but with the first N ones substituted by r other poles. Besides, Corollary 9 states that these new rational functions satisfy a similar recurrence, but with different initial conditions G r , J r , and shifted and rotated parameters γ r+n =τ A λ N +n . Nevertheless, this recurrence does not guarantee the orthogonality because it depends on the orthogonality of the initial conditions G r , J r . Our aim is to complete the hypothesis of Theorem 8 with a minimum number of conditions to ensure the orthogonality of the new rational functions. This is the purpose of the following theorem, which is our main result.
PROOF. Theorem 8 implies that G r+n , J r+n ∈˜ L r+n \ {0}. From (20), (28), (29) and (13) it follows that This, together with (26) and the condition on the function g, gives Next, assuming thatF is a C-function, we also obtain that Further, it follows from (20), (28), (29) and (13) that This, together with (26), the condition on the function g, and the assumption thatF is a C-function, yields . The orthogonality now follows from Theorem 6. 2 The orthogonality properties G r+n ⊥F˜ L r+n−1 and J r+n ⊥ 1/F˜ L r+n−1 imply that the hypothesis of Theorem 10 ensure that, not only G r+n , J r+n ∈˜ L r+n , but G r+n , J r+n ∈˜ L r+n \˜ L r+n−1 too.
Remark 11 ¿From Theorem 5 it follows that, under the same conditions as in Theorem 10, it should hold that Indeed, taking the determinant on both sides of (20), we find for n 0 that wheref Bearing in mind that the left hand side of (30) is in˜ L r+n ·˜ L r+n , it follows that f ∈ L{β 0 } ·˜ L r+n−1 \ {0} for r + n > 0, respectivelyf ∈ C 0 for r + n = 0. Furthermore, taking the superstar conjugate of (30), we obtain that and hence,f (z) =f * (z) ≡d n ∈ R 0 .
A special class of rational functions, the so-called associated rational functions (ARFs), is obtained when τ A = 1 and r = 0. ARFs orthogonal on a subset of the real line are investigated in detail in [8]. Analogously to the case of a subset of the real line, we define the ARFs on the unit circle as follows.
Definition 12 Suppose that the rational functions {φ n } and {ψ n }, with poles among {1/β 1 , . . . , 1/β n }, satisfy a recurrence relation of the form (6). Then, for a given k 0, we call the rational functions φ n\k generated by the recurrence formula k\k ∈ C 0 , the ARFs of {φ n } and {ψ n } of order k.
Note that the subscript '\k' in the definition of the ARFs refers to the fact that the ARFs do not have poles among {1/β 1 , . . . , 1/β k }. In other words, when shifting the recurrence coefficients, the poles are shifted too. Defining L n\k by L (k−1)\k = {0}, L k\k = C, L n\k = L{β k+1 , . . . , β n }, As an application of Theorems 8 and 10, we get an explicit representation of the ARFs and of the function to which they are orthogonal in Theorem 14. But first we need the following lemma.
Lemma 13 Suppose φ k ∈ L k \{0} such that φ k ⊥ F L k−1 , and let ψ k ∈ L k \{0} denote the rational function of the second kind of φ k . It then holds for every z ∈ D that PROOF. First, note that Theorem 6 implies that Moreover, from Theorem 7 it follows that g k (z) + τ ζ k (z)h k (z) = 0 for every τ ∈ T and for every z ∈ D. Therefore, we have that either or G(z) < 0 for every z ∈ D. However, the second option is not possible because for z = β k ∈ D we get G(β k ) = 1 > 0 4 . The statement now follows by multiplying the numerator and denominator in (31) Theorem 14 For n k 0, suppose φ n ∈ L n \ {0} and φ n ⊥ F L n−1 , and let ψ n denote the rational function of the second kind of φ n . Then, there exist constants c n,k ∈ R 0 such that the ARFs φ (k) n\k and ψ (k) n\k are given by n\k ) are orthogonal with respect to the C-function F (k) (respectively 1/F (k) ), given by with F (k) (β k ) = 1. In the special case in which for every n 0 it holds that where d j is the constant defined in Theorem 5, then φ PROOF. First, putβ 0 = β k . Since Φ * k,τ = τ Φ k,τ and Ψ * k,τ = −τ Ψ k,τ , condition (16) is satisfied by A = Ψ k,−1 , B = −Φ k,−1 , C = −Ψ k,1 and D = Φ k,1 with τ A = 1. Further, condition (17) is satisfied too due to Theorem 1.

Theorems 5 and 7 imply that
with g k (z) = 0 and h k (z) = 0 for every z ∈ D, and Hence, conditions (26)-(28) are satisfied too. Further, from (32) and Theorem 5, Consequently, Corollary 9 and Definition 12 show that the rational functions φ n\k are orthogonal with respect to a C-function F (k) , with Note that, forF given by (29), we also have that where the last equality follows from the fact that (φ * k F − ψ * k )(β k ) = 0 (see Theorem 6). Further, from (35)-(36) we get Furthermore, due to Lemma 13. Therefore,F is a C-function, and hence, the equality for F (k) in (33) follows from Theorem 10.
Finally, with c n,k given by (34), it holds for n = k that φ (k) k\k = ψ (k) k\k = 1, while, for n > k, we deduce from (7) and (25) where we have applied (12) in the last equality. Then, the orthonormality is a consequence of (8) in Theorem 4 (2). 2 Based on the previous theorem, the following relations between ARFs of different order can be proved.

Corollary 15
For 0 j k n, let K (j) n,k be defined by .
Then, the following relations hold: where P n\k (z) = P n (z) P k (z) and B n\k (z) = B n (z) B k (z) .
In the special case in which all the involved ARFs are orthonormal, it holds that K PROOF. It suffices to prove the relations for j = 0. Relation (37) follows immediately from (32), with the help of the identity Next, note that (38) is just a reformulation of (37).