Poles and Zeros of Generalized Carathéodory Class Functions

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5-2011

Yael Gilboa College of William and Mary

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Recommended Citation Gilboa, Yael, "Poles and Zeros of Generalized Carathéodory Class Functions" (2011). Undergraduate Honors Theses. Paper 375. https://scholarworks.wm.edu/honorstheses/375

This Honors Thesis is brought to you for free and open access by the Theses, Dissertations, & Master Projects at W&M ScholarWorks. It has been accepted for inclusion in Undergraduate Honors Theses by an authorized administrator of W&M ScholarWorks. For more information, please contact [email protected]. Poles and Zeros of Generalized Carathéodory Class Functions

A thesis submitted in partial fulfillment of the requirement for the degree of Bachelor of Science with Honors in Mathematics from The College of William and Mary

Yael Gilboa

Accepted for (Honors)

Vladimir Bolotnikov, Committee Chair

Ilya Spiktovsky

Leiba Rodman

Julie Agnew

Williamsburg, VA May 2, 2011 POLES AND ZEROS OF GENERALIZED CARATHEODORY´ CLASS FUNCTIONS

YAEL GILBOA

Date: May 2, 2011. 1 2 YAEL GILBOA Contents 1. Introduction 3 2. Schur Class Functions 5 3. Generalized Schur Class Functions 13 4. Classical and Generalized Carathéodory Class Functions 15 5. PolesandZerosofGeneralizedCarathéodoryFunctions 18 6. Future Research and Acknowledgments 28 References 29 POLES AND ZEROS OF GENERALIZED CARATHEODORY´ CLASS FUNCTIONS 3 1. Introduction The goal of this thesis is to establish certain representations for generalized Carathéodory functions in terms of related classical Carathéodory functions. Let denote the Carathéodory class of functions f that are analytic and that have a nonnegativeC real part in the open unit disk D = z : z < 1 . In his papers [10] and [11], Constantin Carathéodory characterized such functions{ | in| terms} of their Taylor coefficients around the origin. Carathéodory class of functions play an important role in the theory of electrical networks. Since the 1920s, there has been extensive research done on related problems. Once Cauer [12] first established the connection between positive functions and electrical circuits, an interest in specifically Carathéodory class functions was prodded on by electrical networks theory. In the 1960s, possible uses for Carathéodory class functions were found in absolute stability theory [17]. An important property of the Carathéodory class is that the kernel f(z)+ f(ζ) Cf (z,ζ)= 1 zζ¯ − is positive on D. During this time, people started to become interested in a wider class of functions such that the kernel Cf (z,ζ) has a finite number of negative squares. This means that the Hermitian matrix n f(z )+ f(z ) i j (1.1) 1 ziz¯j " − #i,j=1 has at most κ negative eigenvalues (counted with multiplicities) for any choice of an integer n and of n points z1,...,zn D at which f is analytic, and it has exactly κ negative eigenvalues for at least one such∈ choice. The positivity of matrices (1.1) is an extremely strong condition that implies, in particular, the analyticity of f even if this analyticity is not an a’priori assumption. However, the uniform bound for the number of negative eigenvalues makes the generalized Carathéodory class κ quite special and closely connected to the classical Carathéodory class . C C The class κ was introduced in the 1960s [4] as the class of functions f that are mero- C morphic in D and such that Cf (z,ζ) has κ negative squares. Extensions of previous results were found to be relevant to generalized Carathéodory class functions and have applications to electrical networks as well. In the contrast to the classical Carathéodory functions, which do not have poles in D and cannot have zeros in D, the functions from the class κ may have up to κ zeros and up to κ poles in D. C

In this thesis, we get a representation formula for κ functions in terms of their poles C and certain related functions. Namely, we will show that every function f κ can be C ∈C 4 YAEL GILBOA written in the form p(z)Φ(z)+ q(z) f(z)= . (1.2) r n n ℓ 2n (z zi) i (1 zzi) i (z zi) i i=1 − − · i=r+1 − where Φ belongs to , p andQ q are polynomials of degreeQ at most 2κ, and zi are the poles or the generalized boundaryC poles of nonpositive type of f. This thesis contains six sections, including the Introduction, and is organized as follows: Section 2 introduces the classical Schur class of complex-valued analytic functions mapping the open unit disk D = z : z 1 intoS the closed unit disk D. Section 3 in- { | |≤ } troduces the generalized Schur class κ. Section 4 introduces the classical and generalized Carathéodory classes, mentioned above.S In Section 5, we derive representation (1.2) as well as an example for the representation of a generalized Carathéodory class function in terms of its poles, zeros and a Carathéodory class function, which inspires our potential future research, briefly outlined in Section 6. POLES AND ZEROS OF GENERALIZED CARATHEODORY´ CLASS FUNCTIONS 5 2. Schur Class Functions The classical Schur class consisting of complex-valued analytic functions mapping the unit disk D = z : z S 1 into the closed unit disk D has been a source of much study and inspiration{ for| over| ≤ a} century now, beginning with the seminal work of Schur (for the original paper of Schur and a survey of some of its impact and applications in signal processing, see [18]). Definition 2.1. A complex-valued function s is called a Schur function if it is analytic on D and satisfies s(z) 1 for every z D. | |≤ ∈ Schur-class functions enjoy many remarkable properties. To present some of them we first recall some needed definitions. Definition 2.2. A matrix A Cn×n is called Hermitian if A = A∗. A Hermitian matrix is called positive semidefinite if∈ all its eigenvalues are nonnegative, and it is called positive definite if all its eigenvalues are positive. Alternatively, we may define positive semidefinite and positive definite matrices as those for which the inner product Ax, x is real and, respectively, nonnegative or positive for every nonzero vector x Cn.h i ∈ On the set of Hermitian n n matrices we introduce Loewner’s partial order: we will say that A B if the matrix×A B is positive semidefinite. We will say that A>B if the matrix A≥ B is positive definite.− − m×n ∗ Definition 2.3. A matrix T C is called contractive if TT Im. ∈ ≤ Lemma 2.4. Let T Cm×n. The following is equivalent: ∈ (1) T is contractive; (2) T ∗ is contractive; (3) Tx x for every x Cn; (4) kT ∗yk ≤ k yk for every y∈ Cm. k k ≤ k k ∈

Im T Proof: The matrix ∗ can be factored in two different ways as follows: T In Im T Im 0 Im 0 Im T ∗ = ∗ ∗ (2.1) T In T In 0 In T T 0 In − ∗ Im T Im TT 0 Im 0 = − ∗ . (2.2) 0 In 0 In T In Assume that T ∗T I; then I T ∗T 0, so the middle matrix in (2.1) is positive ≤ − ≥ Im T Im T semidefinite. Then ∗ is positive semidefinite. Since ∗ is invertible, it T In T In ∗ follows from (2.2) that the middle matrix is positive semidefinite. Then Im TT 0 or ∗ − ≥ TT Im. By reversing the arguments, we get the equivalence of (1) and (2). ≤ 6 YAEL GILBOA Assume that Tx x for every x. Then Tx 2 x 2, which can be written as || ||≤k k || || ≤ || || 0 x 2 Tx 2 = x,x Tx,Tx = x∗x x∗T ∗Tx = x∗(I T ∗T )x. ≤ || || − || || h i−h i − − So x∗(I T ∗T )x 0 for every x, which means that (I T ∗T ) is positive semidefinite and − ≥ − T ∗T I. We have proved (3) (2). By reversing the arguments, we get (2) (3). ≤ ⇒ ⇒ Assume that T ∗y y for every y. Then T ∗y 2 y 2, which can be written as || ||≤k k || || ≤ || || 0 y 2 T ∗y 2 = y,y T ∗y,T ∗y = y∗y y∗TT ∗y = y∗(I TT ∗)y. ≤ || || − || || h i−h i − − So y∗(I TT ∗)y 0 for every y, which means that (I TT ∗) is positive semidefinite and TT ∗ − I. We≥ have proved (4) (1). By reversing− the arguments, we get (1) (4). ≤ ⇒ ⇒

To present a remarkable property of Schur-class functions, let us recall the Hardy space H2 consisting of all functions f with square-summable Taylor coefficients: ∞ ∞ 2 k 2 H = f(z)= fkz : fk < . | | ∞ ( k k ) X=0 X=0 This space can be endowed with inner product ∞ ∞ ∞ k k f,g 2 = fkg for f(z)= fkz , g(z)= gkz . (2.3) h iH k k k k X=0 X=0 X=0 Then for every f H2 we can define the norm ∈ 1 2 1 ∞ 2 2 f 2 = f,f 2 = fk . k kH h iH | | k ! X=0 1 2 Lemma 2.5. The function ka(z) = belongs to H for every fixed a D and 1 za ∈ 1 − ka H2 = . k k 1 a 2 −| | Proof: pThe function ka(z) can be represented by the Taylor expansion: ∞ ∞ n n n ka(z)= (za) = a z (2.4) n=0 n=0 X X Using the definition of the norm in H2 and the Taylor coefficients of an, one can see ∞ 2 2n 1 ka 2 = ka, ka 2 = a = . k kH h iH | | 1 a 2 n=0 X −| | 1 So, ka H2 = . k k 1 a 2 −| | p POLES AND ZEROS OF GENERALIZED CARATHEODORY´ CLASS FUNCTIONS 7

The functions ka have the following reproducing property: the inner product of ka with any function f H2 recovers the value of f at a. ∈ Lemma 2.6. For every f H2 and any a D, we have ∈ ∈ f, ka 2 = f(a). (2.5) h iH

Proof: Using deﬁnition (2.3) of the inner product of H2 and the Taylor expansion for ka(z) (2.4), we have ∞ n f, ka 2 = fna = f(a). h iH n=0 X Let us deﬁne: 1 ∂n zn k (z)= k (z)= . (2.6) a,n n! ∂a a (1 za)n+1 −

2 2 Lemma 2.7. For every f H and any a D, we have ka,n H as a rational function with no poles in D and ∈ ∈ ∈ f (n)(a) f, ka,n 2 = . (2.7) h iH n!

1 Proof: The Taylor series for f(z)= is (1 za)n+1 − ∞ n + k f(z)= akzk. k k X=0 n Thus, the Taylor series for ka,n(z)= z f(z) is ∞ n + k k (z)= akzn+k. (2.8) a,n k k X=0 2 Using the deﬁnition (2.3) of inner product of H , the Taylor expansion (2.8) for ka,n(z), and (2.6) we have

∞ (n) n + k k f (a) f, ka,n 2 = fn k a = . h iH + k n! k X=0

It was shown by Pierre Fatou [14] that for any Schur-class function s, the radial boundary limit s(eit)= lim s(reit) r→1− 8 YAEL GILBOA exists for almost all t [0, 2π). It is clear that this boundary limit does not exceed one in modulus whenever it∈ exists. Fatou also showed that if f belongs to H2, then for almost all t [0, 2π), the radial boundary limit ∈ f(eit)= lim f(reit) r→1− exists and moreover,

∞ 2π 2 2 1 it 2 f 2 := fk = f(e ) dt. (2.9) k kH | | 2π | | k 0 X=0 Z Every Schur-class function s deﬁnes the linear map Ms : f s f. Assuming that f belongs to H2 let us compare∈ S the norms f and sf using the→ formula· (2.9) and the fact that s(eit) 1 for almost all t [0, 2πk):k k k | |≤ ∈ 2π 2 1 it it 2 sf 2 = s(e )f(e ) dt k kH 2π | | Z0 1 2π = s(eit) 2 f(eit) 2dt 2π 0 | | ·| | Z 2π 1 it 2 2 f(e ) dt = f 2 . ≤ 2π | | k kH Z0 2 The latter calculation shows that Msf belongs to H whenever f does and moreover, that Ms does not increase the norm. Thus, Ms is a contraction by Lemma 2.4. Therefore the ∗ adjoint map Ms is also contractive by Lemma 2.4.

n n 2 Lemma 2.8. Let s(z)= snz and f(z)= fnz H . Then ∈S ∈

P ∞ ∞ P ∗ n Ms f = skfn+k z . (2.10) n=0 k ! X X=0

∗ 2 Proof: The function H = Ms f belongs to H . Let us expand H in Taylor series

∞ H(n)(0) H(z)= zn n! n=0 X and show that H(n)(0) ∞ = s f . n! k n+k k X=0 Indeed, by (2.7), POLES AND ZEROS OF GENERALIZED CARATHEODORY´ CLASS FUNCTIONS 9

(n) H (0) n = H, k ,n = H,z n! h 0 i h i ∗ n n n = M f,z = f, Msz = f,sz h s i h i h i ∞ ∞ k k = fkz , sk nz h + i k=0 k=0 ∞X X

= fk+nsk. k X=0 The formula (2.10) takes a particularly simple form in the case where f equals the elementary kernel ka. 1 Corollary 2.9. Let s and let ka(z)= . Then ∈S 1 za − ∗ Ms ka = s(a)ka. (2.11)

n ∗ Proof: Let us apply (2.10) to f = ka. So fn = a and for H = Ms ka, we have

(n) ∞ ∞ ∞ H (0) k+n n k n = skfk n = a sk = a a sk = a s(a). n! + · k k k X=0 X=0 X=0 Therefore,

∞ (n) ∞ ∞ ∗ H (0) n n n n n M ka = z = a s(a)z = s(a) a z = s(a)ka. s n! · n=0 n=0 n=0 X X X Remark 2.10. Equality (2.11) means that elementary kernels ka are eigenvectors of the ∗ operator Ms while the complex-conjugate values s(a) are the corresponding eigenvalues. Let us consider the function n cj g(z)= c1ka1 (z)+ ... + cnkan (z)= (2.12) 1 zaj j=1 X − where a1,...,an are any n distinct points in D and c1,...,cn are arbitrary ﬁxed complex numbers. This function belongs to H2 since H2 is a vector space. ∗ Since the operator Ms is linear, we have from (2.11), n n n ∗ ∗ ∗ Ms g = Ms cikai = ciMs kai = cis(ai)kai . (2.13) i=1 i=1 i=1 X X X 10 YAEL GILBOA We now use the reproducing property (2.5) to get

n 2 2 g 2 = cika 2 k kH k i kH i=1 Xn n

= cjkaj , cikai * j=1 i=1 + nXn X

= cicjkaj (ai) i=1 j=1 Xn Xn 1 = cicj . (2.14) 1 aiaj i=1 j=1 X X − Quite similarly we get from (2.13)

∗ 2 ∗ ∗ 2 Ms g H2 = Ms g, Ms g H k k h n i n

= cis(ai)ka , cjs(aj)ka h i j i i=1 j=1 Xn n X

= cicjs(ai)s(aj)kaj (ai) i=1 j=1 Xn Xn s(ai)s(aj) = cicj . (2.15) 1 aiaj i=1 j=1 X X − ∗ If s is a Schur-class function, the map Ms is a contraction. Therefore, Ms g g , for every g H2. In particular, for the function g of the form (2.12) we havek k ≤ k k ∈ M ∗g 2 g 2, k s k ≤ k k which on account of (2.14) and (2.15) can be written as

n n n n s(ai)s(aj) 1 cicj cicj , 1 aiaj ≤ 1 aiaj i=1 j=1 i=1 j=1 X X − X X − which is the same as n n 1 s(ai)s(aj) cicj − 0. (2.16) 1 aiaj ≥ i=1 j=1 X X − Let us consider the matrix n s 1 s(ai)s(aj) P (a1,...,an)= − . (2.17) 1 aiaj " − #i,j=1 POLES AND ZEROS OF GENERALIZED CARATHEODORY´ CLASS FUNCTIONS 11 Then condition (2.16) means that

c1 s . c1 ... cn P (a1,...,an) . 0   ≥ c n ∗ s   for every choice of c1,...,cn. In other words, x P (a1,...,an)x 0 for every vector- n s ≥ column from C , which, in turn, means that the matrix P (a1,...,an) is positive semi- deﬁnite. We thus proved that if s is a Schur-class function, then for every choice of a s positive integer n and for any n distinct points in D, the matrix P (a1,...,an) of the form (2.17) is positive semideﬁnite.

The converse statement is of special interest. As Alan Hindmarsh showed in [19], if s is s a function defined on D and such that the matrices P (a1,a2,a3) are positive semidefinite for any triple of distinct points a1,a2,a3 in D, then s is a Schur-class function. The boundedness property s(z) 1 is immediate even from positivity of 1 1 matrices. A remarkable part of Hindmarsh’s| | ≤ result is that positivity of all 3 3 matrices× implies that × s is analytic and therefore, the matrices Pn are positive semidefinite for all integers n 1. ≥

Deﬁnition 2.11. Let Ω be a domain in C. A function K(z,ζ) is called a positive kernel on Ω Ω if it is deﬁned for all z,ζ Ω and × ∈ r

cicjK(zi,zj) 0 ≥ i,j=1 X for every choice of a positive integer r and of points z1,...,zn Ω or equivalently, if the r ∈ matrix [K(zi,zj)]i,j=1 is positive semidefinite for any such choice. Our previous result just means that if s is a Schur-class function, then the kernel 1 s(z)s(ζ) Ks(z)= − (2.18) 1 zζ − is positive on D D. We summarize the above discussion in the following theorem. × Theorem 2.12. Let s be a complex-valued function analytic on D. Then the following are equivalent: (1) s belongs to , i.e., s(z) 1 for every z D. S | |≤ ∈ (2) The kernel Ks(z,ζ) defined in (2.18) is positive on D D. × (3) The matrix Ps(a1,...,an) is positive semidefinite for any choice of a ,...,an D. 1 ∈

In what follows, we will refer to the matrix Ps(a1,...,an) as to the Pick matrix of the function s based on points a ,...,an D. 1 ∈ 12 YAEL GILBOA We now present simple examples of Schur-class functions. The function z a ba(z)= − (a D), (2.19) 1 za¯ ∈ − is called a Blaschke factor. This is a linear fractional function with its only zero at z = a and its only pole at z =1/a. Let us show that ba belongs to . Indeed, S z a 2 1 za¯ 2 z a 2 1 − = | − | −| − | − 1 za¯ 1 za¯ 2 − | − | (1 za¯)(1 za¯ ) (z a)(¯z a¯) = − − − − − 1 za¯ 2 | − | (1 a 2)(1 z 2) = −| | −| | , 1 za¯ 2 | − | which implies that ba(z) < 1 if z D and that ba(z) = 1 if z = 1. The product of finitely many Blaschke| factors| ∈ | | | | n n z aj f(z)= baj (z)= − (2.20) 1 za¯j j=1 j=1 Y Y − where ai’s are not necessarily distinct, is called a finite Blaschke product. This function is rational of degree n and has zeros at ai and poles at 1/ai. Since the modulus of the product of two complex numbers is equal to the product of moduli, it follows that f(z) < 1 if z D and that f(z) = 1 if z = 1 and in particular, that any finite Blaschke| product| belongs∈ to the Schur| | class. The| | set of all Blaschke products of degree k will be denoted by k. By we denote the set of all constant unimodular functions. B B0 The class k admits the following characterization in terms of Pick matrices (see e.g., [6] for a proof).B

Theorem 2.13. A function f belongs in k if and only if for every n 1 and any B ≥ choice of a ,...,an D, the Pick matrix Ps(a ,...,an) is positive semideﬁnite and 1 ∈ 1 rankPs(a ,...,an) = min k,n . 1 { } POLES AND ZEROS OF GENERALIZED CARATHEODORY´ CLASS FUNCTIONS 13 3. Generalized Schur Class Functions

For a ﬁxed integer κ 0, we deﬁne the generalized Schur class κ as the set of all meromorphic functions of≥ the form S s(z) f(z)= , (3.1) b(z)

where s and b κ do not have common zeros (in particular, = ). Originally, ∈ S ∈ B S0 S the classes κ appeared implicitly in approximation theory [2] and in interpolation theory S [20]. Later V. Adamyan, D. Arov and M.G. Krein [1] discovered deep connections of κ S with spectral theory of Hankel operators. The thorough study of κ classes is presented in [15, 16], while more recent advances can be found in the monographS [3].

Formula (3.1) is called the Krein-Langer representation of a generalized Schur function f; the entries s and b are determined by f uniquely up to a unimodular constant. Since every finite Blaschke product is a rational function analytic on a neighborhood of D, the function f of the form (3.1) admits radial boundary limits whenever the denominator s does, that is, almost everywhere on T. Since b(eit) = 1 for every t [0, 2π), it follows∈ S that | | ∈ f(eit) = s(eit) 1. | | | |≤ Therefore, via radial boundary limits, the κ functions can be identified with the functions defined almost everywhere on T, which areS bounded by one in modulus (that is, with the functions from the unit ball of L∞(T)) and which admit the meromorphic continuation inside the unit disk with total pole multiplicity equal to κ. On the other hand, the κ functions f can be characterized as meromorphic functions on D for which all the Sassociated Pick matrices n s 1 s(ai)s(aj) P (a1,...,an)= − (3.2) 1 aiaj " − #i,j=1 have not more than κ negative eigenvalues (counted with multiplicities) and some of these matrices have exactly κ negative eigenvalues. Of course, the points a1,...,an are chosen in the domain of definition of f. It was shown in [9] that for an f κ, there always exist κ κ Pick matrices that have κ negative eigenvalues. ∈S × Definition 3.1. Let Ω be a domain in C. The kernel K(z,ζ) is said to have κ negative squares – in notation, sq−K(z,ζ)= κ if (1) It is defined for all z,ζ Ω; ∈ (2) It is Hermitian in the sense that K(z,ζ)= K(ζ,z) for all z,ζ Ω; ∈ (3) For every choice of a positive integer r and of points z1,...,zn Ω, the Hermitian r ∈ matrix [K(zi,zj)]i,j=1 has at most κ negative eigenvalues and it has exactly κ negative eigenvalues for at least one such choice. 14 YAEL GILBOA

This deﬁnition and the preceding discussion imply that κ can be characterized as the class of meromorphic functions f such that the associatedS kernel 1 f(z)f(ζ) Kf (z,ζ) := − (3.3) 1 zζ¯ − has κ negative squares on ρ(f), the domain of analyticity of f: sq−(Kf )= κ. We conclude this section with a result concerning Hermitian kernels. Theorem 3.2. Let K(z,ζ) be a Hermitian kernel on Ω Ω and let A(z) be a function × such that A(z) = 0 for all z Ω. Then the kernel K(z,ζ) = A(z)K(z,ζ)A(ζ) has the same number of6 negative squares∈ as K(z,ζ): e sq−K(z,ζ)=sq−A(z)K(z,ζ)A(ζ)=sq−K(z,ζ).

e n ˜ Proof: Choose z1,...,zn Ω and consider the matrices P = [K(zi,zj)]i,j=1 and P = n ∈ n K˜ (zi,zj) = A(zi)K(zi,zj)A(zj) . i,j=1 i,j=1 h Then iti is easilyh seen that P˜ = CPCi∗, where

A(z1) 0 C = ... .   0 A(zn) Since A(z) = 0, then C is invertible. If the kernel K has m negative squares, then the matrix P 6has at most m negative eigenvalues, and for some choice of points, it has exactly m. Since P˜ has the same inertia as P (meaning they have the same numbers of positive, negative and zero eigenvalues, including multiplicities), that means that P˜ has at most m negative eigenvalues, and for some choice of points, it has exactly m. So, the kernel K˜ has exactly m negative squares. POLES AND ZEROS OF GENERALIZED CARATHEODORY´ CLASS FUNCTIONS 15 4. Classical and Generalized Caratheodory´ Class Functions Deﬁne the Carath´eodory class to be a class of functions f that are analytic and have a nonnegative real part in D. ItC will be useful for us to be able to switch between the classes and with relative ease. For this reason, the following theorems have been formulated,C whichS will be applied to functions in and throughout what follows in this thesis: C S Theorem 4.1. A function f belongs to if and only if it can be represented as f(z) = 1+ s(z) C f(z) 1 for some s in . Clearly, this s equals s(z)= − . 1 s(z) S f(z)+1 − f 1 Proof: For s = − , we have f +1

(f(z) 1)(f(ζ) 1) 1 s(z)s(ζ)=1 − − − − (f(z)+1)(f(ζ)+1) (f(z)+1)(f(ζ)+1) (f(z) 1)(f(ζ) 1) = − − − (f(z)+1)(f(ζ)+1) (4.1) f(z)f(ζ)+ f(ζ)+ f(z)+1 f(z)f(ζ)+ f(z)+ f(ζ) 1 = − − (f(z)+1)(f(ζ)+1) 2f(z)+2f(ζ) = . (f(z)+1)(f(ζ)+1) In particular, for z = ζ, we have 2f(z)+2f(z) 1 s(z) 2 = 0. −| | f(z)+1 2 ≥ | | The inequality is true because f belongs to the Carath´eodory class. Since Ref(z) 0, f(z) 1 ≥ it follows that f(z)+1 = 0 for all z in D. Then s(z)= − is analytic on D. Further, 6 f(z)+1 since s(z) 1, then s is indeed in the Schur class. | |≤ Note also that if f , then the kernel ∈C f(z)+ f(ζ) Cf (z,ζ)= 1 zζ¯ − is positive on D. Indeed, it follows from (4.1) that

1 s(z)s(ζ) √2 √2 Ks(z,ζ)= − = Cf (z,ζ) (4.2) 1 zζ¯ f(z)+1 f(ζ)+1 − 16 YAEL GILBOA √2 where s is deﬁned as in Theorem 4.1. Since A(z) = = 0, the kernels Ks and f(z)+1 6 Cf have the same number of negative squares. Since f , we have s . Then Ks is ∈ C ∈ S positive on D, and so Cf is positive on D.

Deﬁne κ to be a class of functions f that are meromorphic in D and such that Cf (z,ζ) has κ negativeC squares. We need the following well-known result.

Lemma 4.2. For any n distinct points z ,...,zn D, the matrix 1 ∈ 1 n F = 1 zizj − i,j=1 is positive deﬁnite. 1 Proof: The functions kzj (z)= , j =1,...,n are linearly independent functions 1 zzj in H2. Therefore, the Gram matrix− consisting of all their inner products

kz1 , kz1 kz1 , kzn n h i ··· h i . .. . kzj , kzi = . . . h i i,j=1   kz , kz1 kz , kz h n i ··· h n n i is positive deﬁnite. Since   1 1 1 , = 1 zzj 1 zzi 2 1 zizj − − H − by Lemma 2.6, the Gram matrix equals F .

The next theorem provides a connection between functions in the two classes κ and S κ. C 1+ s Theorem 4.3. A function f belongs to κ if and only if it can be represented as f = C 1 s for some s Sκ. − ∈ f 1 Proof: Given f κ, define s = − . First let us show that this s is well-defined, ∈ C f +1 meaning that f 1. Indeed, if f(z) 1, then the kernel Cf has infinitely many 6≡ − ≡ − negative squares. To show this, take z1,...,zn D. ∈ n n n 2 1 P =[Cf (zi,zj)]i,j=1 = = 2 −1 zizj − 1 zizj − i,j=1 − i,j=1 has n negative eigenvalues. If f(z) = 1 for all z, then P would have infinitely many negative eigenvalues as you increase the− size of the matrix. If f and s are related as above, then the kernels Ks and Cf are related as in (4.2). Thus, sq−Ks = κ by Theorem 3.2. Therefore, s κ. The reverse argument is also true by Theorem 3.2. ∈S POLES AND ZEROS OF GENERALIZED CARATHEODORY´ CLASS FUNCTIONS 17

Corollary 4.4. Every function f κ admits a representation ∈C b + g f = (4.3) b g − where g and b κ have no common zeros. ∈S ∈B Proof: Combining (4.3) and the Krein-Langer representation of a generalized Schur 1+ s function from (3.1), we define f = k and have the equality 1 sk − 1+ s 1+ g b + g f = = b = , 1 s 1 g b g − − b − where s . ∈S 18 YAEL GILBOA 5. Poles and Zeros of Generalized Caratheodory´ Functions We now take the advantage of the representation formula (4.3) to study poles and zeros of generalized Carathéodory functions. We first note that for a finite Blaschke product b κ and a Schur-class function g , the function h = g b may have at most κ zeros ∈B ∈S − inside D counted with multiplicities. Let z1,...,zr be all such zeros and let n1,...,nr be their respective multiplicities. Then (j) (j) g (zi)= b (zi) for j =0,...,ni 1; i =1,...,r. (5.1) − It was shown in [7] that if n +...+nr is less than κ, then the function h = g b must have 1 − certain zeros at the boundary of D. More precisely, there exist points zr ,...,zℓ T +1 ∈ with respective multiplicities nr+1,...,nℓ such that the nontangential boundary limits (j) hij = lim h (z) exist for j =0,..., 2ni 1 and are such that hij = 0 for j =0,..., 2ni 2 z→zi − − and ni 2ni−1 ( 1) z b(zi)hi, n 0 for i = r +1,...,ℓ. − i 2 i−1 ≥ Since b is analytic on T, the latter is equivalent to the existence of the nontangential boundary limits lim g(j)(z) and the relations z→zi (j) (j) lim g (z)= b (zi) for j =0,..., 2ni 2; i = r +1,...,ℓ (5.2) z→zi − and

ni 2ni−1 (2ni−1) (2ni−1) ( 1) zi b(zi)(lim g (z) b (zi)) 0 for i = r +1,...,ℓ. (5.3) − z→zi − ≥

An interesting thing to note here is that n1 + ... + nr + ... + nℓ = κ.

Given a function f with a representation (4.3), let us say that zi is a generalized pole of f of nonpositive type of multiplicity 2ni if conditions (5.2) and (5.3) are satisﬁed. It is clear that if conditions (5.1) are satisﬁed at z ,...,zr D, then zi is a usual pole of f of 1 ∈ order ni.

Given b κ and g , let z1,...,zr D and zr+1,...,zℓ T be the zeros of the function g ∈b Bdescribed∈ as S above. Let us introduce∈ the numbers∈ − (j) b (zi) j =0,...,ni 1, if i 1,...,r , cij := for − ∈ { } (5.4) j! j =0,..., 2ni 1, if i r +1,...,ℓ − ∈ { } and let us consider the following interpolation problem: IP: ﬁnd all g such that ∈S (j) g (zi)= j! cij for j =0,...,ni 1; i =1,...,r, (5.5) · − (j) (j) g (zi) := lim g (z)= j! cij (5.6) z→zi · for j =0,..., 2ni 2; i = r +1,...,ℓ, and − ni−1 2ni−1 (2ni−1) ( 1) z ci, g (zi) (2ni 1)! ci, n 0 (5.7) − i 0 − − · 2 i−1 ≤ POLES AND ZEROS OF GENERALIZED CARATHEODORY´ CLASS FUNCTIONS 19 for i = r +1,...,ℓ. This problem is well known (see e.g., [5, 8]) and its solution is recalled below.

Let us associate with the given tuples z =(z1,...,zℓ) and n =(n1,...,nℓ) the matrices

Jn1 (z1) En1 . T = ... and E = . , (5.8)    .  Jnk (zk) Enk     where we denote by Jn(z) and En respectively the n n Jordan block with z C on the main diagonal and the vector of the length n where× the ﬁrst coordinate equals∈ one and other coordinates equal zero: z 0 ... 0 1 .. . 0  1 z . .  Jn(z)= . . , En =  .  . (5.9) .. .. 0 .    0 1 z   0          The numbers cij from (5.5)–(5.7) are arranged in the column

C1,n1 ci,0 C = . , where C = . . (5.10)  .  i,ni  .  Ck,nk ci,ni−1     Observe that Cn contains all the numbers cij but ci,ni ,...,ci,2ni−1 for all i = r +1,...,ℓ. These extra numbers are used to deﬁne the structured matrix Pii as follows:

ci,1 ci,2 ci,ni ··· ci,0 ... ci,ni−1 ci, ci, ci,n 2 3 ··· i+1 .. . Pii =  . . .  Ψni (zi) . . (5.11) . . .   0 c  c c ... c  i,0  i,ni i,ni+1 i,2ci−1      (i = r +1,...,ℓ) where the ﬁrst factor is a Hankel matrix, the third factor is an upper n1−1 triangular Toeplitz matrix and where Ψni (z1)=[Ψjℓ]j,ℓ=0 is the upper triangular matrix n − 1 2 3 ni−1 i ni zi zi zi ( 1) 0 zi − ··· − n − 1  0 z3 2z4 ( 1)ni−1 i zn1+1  − i i ··· − 1 i . n − 1 Ψ  . 5 ni−1 i ni+2  ni (zi)=  . zi ( 1) 2 zi  , (5.12)  ··· −   . . .   . .. .   n − 1   0 0 ( 1)ni−1 i z2ni−1   ni − 1 i   ··· ··· −  with entries   ℓ ℓ ℓ+j+1 Ψjℓ =( 1) z , 0 j ℓ ni 1. (5.13) − j i ≤ ≤ ≤ − 20 YAEL GILBOA

The matrix Pii is not Hermitian in general. However, if the numbers cij are the Taylor coefficients of a finite Blaschke product at a boundary point, then the matrix Pii is not only Hermitian, but also positive semidefinite. This was proved in [8]. Moreover, whenever Pii is Hermitian and ci,0 is unimodular, the matrix Pii satisfies the following Stein identity: ∗ ∗ ∗ Pii Jn (zi)PiiJn (zi) = En E Cn C (5.14) − i i i ni − i ni where Jn (zi), En and Cn are given in (5.9), (5.10). Furthermore, if i = j or if i = i i i 6 j 1,...,r , we define the ni nj matrix Pij as the unique matrix satisfying the Stein identity∈ { } × ∗ ∗ ∗ Pij Jn (zi)PijJn (zj) = En E Cn C . (5.15) − i j i nj − i nj

The latter equality indeed has a unique solution since the only eigenvalues of Jni (zi) and

Jnj (zj) are zi and zj respectively and zizj = 1. In this case, the entries of Pij can be found recursively (see e.g., [5]): 6

min{ℓ,r} r−s ℓ−s (ℓ + r s)! zi z¯j [Pij]ℓ,r = − ℓ+r−s+1 (5.16) (ℓ s)!s!(r s)! (1 ziz¯j) s=0 X − − − ℓ r min{α,β} β−s α−s (α + β s)! zi z¯j ci,ℓ−αcj,r−β − α+β−s+1 . − (α s)!s!(β s)! (1 ziz¯j) α=0 β s=0 X X=0 X − − − It now follows that the block matrix P given by ℓ P =[Pij]i,j=1 (5.17) with the blocks defined in (5.11) and (5.17) is Hermitian and satisfies the Stein identity P TT ∗ = EE∗ CC∗. (5.18) − − where T , E and C are given in (5.9), (5.10). Moreover, since the numbers cij are the Taylor coefficients of a finite Blaschke product of degree κ, the matrix P is positive definite. With all these ingredients in hands, we introduce the 2 2 matrix function × θ (z) θ (z) Θ(z) = 11 12 (5.19) θ (z) θ (z) 21 22 1 0 E∗ = +(z µ) (I zT ∗)−1P −1(µI T )−1 E C , 0 1 − C∗ − − − where µ is an arbitrary point in T x1,...,xℓ . The following theorem (see [5], [8] for the proof) describes all functions g \{ satisfying} conditions (5.5)–(5.7). ∈S Theorem 5.1. A function g belongs to the Schur class and satisfies conditions (5.5)– (5.7) if and only if it is of the form S θ + θ g = 11E 12 (5.20) θ + θ 21E 22 for some . Moreover, if m, then g m κ. E∈S E∈B ∈B + POLES AND ZEROS OF GENERALIZED CARATHEODORY´ CLASS FUNCTIONS 21

The functions θij are rational and of degree at most κ. Let us introduce the signature 1 0 matrix J = . A straightforward calculation based solely on the identity (5.18) 0 1 shows that for every− z,ζ at which Θ is analytic, E∗ J Θ(z)JΘ(ζ)∗ = (1 zζ¯) (I zT ∗)−1P −1(I ζT¯ )−1 E C . − − C∗ − − In particular, Θ(z)JΘ(ζ)∗ J if z < 1 ≤ | | and Θ(z)JΘ(ζ)∗ = J if z =1. | | Another calculation based on the identity (5.18) gives

ℓ ni (z zi)(1 µzi) det Θ(z)= − − . (5.21) (1 zzi)(µ zi) i=1 Y − − Indeed, since det(I + AB) = det(I + BA), we have E∗ detΘ(z) = det I +(z µ) (I zT ∗)−1P −1(µI T )−1 E C − C∗ − − − E∗ = det I +(z µ)(I zT ∗)−1P −1(µI T )−1 E C − − − − C∗ = det (I zT ∗)−1P −1(µI T )−1 − − det((µI T )P (I zT ∗)+(z µ)(EE∗ CC∗)) · − − − − = det (I zT ∗)−1P −1(µI T )−1 − − det((µI T )P (I zT ∗)+(z µ)(P TPT ∗)) · − − − − = det (I zT ∗)−1P −1(µI T )−1 − − det(zP T P µzPT ∗ + µTPT ∗) · − − = det (I zT ∗)−1P −1(µI T )−1 det((zI T )P (I µT ∗)) − − · − − = det (I zT ∗)−1(µI T )−1(zI T )(I µT ∗) . − − − − Due to the special structure (5.8), (5.9) of T ,

ℓ ℓ ni det(zI T )= det(zIn Jn (zi)) = (z zi) − i − i − i=1 i=1 Y Y and quite similarly, ℓ ∗ ni det(I zT )= (1 zzi) . − − i=1 Y 22 YAEL GILBOA

The three latter formulas imply (5.21). Observe that since zr+1,...,zℓ fall on the unit circle, we have

z zi zi(zzi 1) − = − = zi 1 zzi 1 zzi − − − for i = r +1,...,ℓ. Thus,

r ni z zi det Θ(z)= γ − . (5.22) · 1 zzi i=1 Y − where

ℓ ni ℓ 1 µzi ni γ = − ( zi) (5.23) µ zi · − i=1 i=r+1 Y − Y is a unimodular number, since

ℓ ni ℓ ni ℓ ni 1 µzi µ zi µ zi − = µ − = µ ni − =1. µ zi · µ zi | | ·µ zi i=1 i=1 i=1 Y − Y − Y −

Theorem 5.2. Let f be a function in κ with poles z ,...,z r D of respective orders C 1 ∈ n1,...,nr and with generalized boundary poles zr+1,...,zℓ of nonpositive type of respective orders 2nr+1,..., 2nℓ. Then there exist polynomials p and q of degree at most 2κ and a Carath´eodory-class function Φ such that ∈C p(z)Φ(z)+ q(z) f(z)= . (5.24) r n n ℓ 2n (z zi) i (1 zzi) i (z zi) i i=1 − − · i=r+1 − Q Q

Proof: If f and b κ are the functions from the representation (4.3), they are solutions to the∈ problem S IP∈ Bby construction. Therefore, g is of the form (5.20) for some and b admits a similar representation E∈S θ b + θ b = 11 12 (5.25) θ21b + θ22 e with b . By Theorem 5.1, b 0 that is beis a unimodular constant. Evaluating (5.25) ∈S ∈B at z = µ gives b(µ) = b. Since we may assume without loss of generality that b(µ) = 1, we gete e e e θ + θ b = 11 12 . (5.26) θ21 + θ22 POLES AND ZEROS OF GENERALIZED CARATHEODORY´ CLASS FUNCTIONS 23 Using (5.20), (5.26) and Corollary 4.4 , we have θ + θ θ + θ 11 12 + 11E 12 b + g θ + θ θ + θ = 21 22 21E 22 b g θ11 + θ12 θ11 + θ12 − E θ + θ − θ + θ 21 22 21E 22 (θ + θ )(θ + θ )+(θ + θ )(θ + θ ) (5.27) = 11E 12 21 22 11 12 21E 22 (θ + θ )(θ + θ ) (θ + θ )(θ + θ ) 11E 12 21 22 − 11 12 21E 22 2 θ θ +2θ θ +( + 1)(θ θ + θ θ ) = E 11 21 12 22 E 11 22 12 21 . (θ θ θ θ )(1 ) 11 22 − 12 21 −E 1+ Φ 1 Define a function Φ using Theorem 4.1 to be Φ = E . Thus, = − . So, ∈ C 1 E Φ+1 2 2Φ −E 1 = and 1+ = . Substituting the latter equalities into (5.27), we have −E Φ+1 E Φ+1 b + g 2 θ θ +2θ θ +( + 1)(θ θ + θ θ ) = E 11 21 12 22 E 11 22 12 21 b g detΘ (1 ) − · −E Φ 1 Φ − θ11θ21 + θ12θ22 + (θ11θ22 + θ12θ21) = Φ+1 Φ+1 1 detΘ · Φ+1 (Φ 1)θ θ +(Φ+1)θ θ + Φ(θ θ + θ θ ) = − 11 21 12 22 11 22 12 21 (5.28) detΘ (Φ 1)θ θ +(Φ+1)θ θ + (Φ)(θ θ + θ θ ) = − 11 21 12 22 11 22 12 21 θ θ θ θ 11 22 − 12 21 Φ(θ θ + θ θ + θ θ + θ θ ) θ θ + θ θ = 11 21 12 22 11 22 12 21 − 11 21 12 22 detΘ (θ + θ )(θ + θ )Φ + θ θ θ θ = 11 12 21 22 12 22 − 11 21 . detΘ Let us look at the formula (5.19). It is clear due to the block-diagonal structure (5.8) of T that ∗ −1 (I zJn1 (z ) ) 0 − 1 (I zT ∗)−1 = ... −  ∗ −1 0 (I zJnℓ (zℓ) )  −  and since 1 0 ... 0 1−zz¯i 1 1 .. . 2 . . ∗ −1  (1−zz¯i) 1−zz¯i  (I zJn1 (z ) ) = − 1 .. ..  . . 0   1 1 1   ni 2   (1−zz¯i) (1−zz¯i) 1−zz¯i    24 YAEL GILBOA (see (5.9)), it follows that the function ℓ ni ∗ −1 (1 zz¯i) (I zT ) − · − i=1 Y is a matrix polynomial. Now it follows from (5.19) that the functions ℓ ni θij(z)= (1 zz¯i) θij(z). i, j =1, 2, − · i=1 Y are polynomials. Let use set p = γ(θ + θ )(θ + θ ) and q = γ(θ θ θ θ ). 11 12 21 22 12 22 − 11 21 where γ is the unimodular number defined in (5.23). If we multiply the denominator on the right hand side of (5.28)e e by thee functione e e e e ℓ 2ni γ (1 zz¯i) , − i=1 Y we get the expression equal the denominator in (5.24). Multiplying the numerator on the right hand side of (5.28) by the same function gives pΦ+ q by the very definition of p and q. This completes the proof of the theorem. Theorem 5.2 elaborated the poles of a generalized Carathéodory function. The zeros of f can be elaborated in much the same way. All zeros of such an f are the points ζ ,...,ζα D at which the functions g and b κ coincide up to certain order: 1 ∈ ∈S − ∈B (j) (j) g (ζi)= b (ζi) for j =0,...,mi 1; i =1,...,α. (5.29) − − Due to a result from [7], if m +...+mα is less than κ, then there exist points ζα ,...,ζβ 1 +1 ∈ T with such that the nontangential boundary limits lim g(j)(z) exist and satisfy z→ζi (j) (j) lim g (z)= b (ζi) for j =0,..., 2mi 2; i = α +1,...,β (5.30) z→ζi − − and

mi 2mi−1 (2mi−1) (2mi−1) ( 1) zi b(ζi)(lim g (z)+ b (ζi)) 0 for i = α +1,...,β. (5.31) − z→ζi ≤

Once again, m1 + ... + mβ = κ. We may think of (5.29)–(5.31) as of an interpolation (j) problem similar to IP but with diﬀerent data dij = b (ζi)/j!. We may follow the above strategy to get a representation of f emphasizing− its zeros. However, there are several indications that these two descriptions can be combined. We plan to pursue our investigations in this direction. Here we present a simple case where f is of the form g(z)+ z f(z)= (5.32) z g(z) − and has one pole and one zero in D. Thus, we assume that g(z )= z and g(z )= z . (5.33) 1 1 2 − 2 POLES AND ZEROS OF GENERALIZED CARATHEODORY´ CLASS FUNCTIONS 25 Since g is a Schur-class function,

2 1+ z1z¯2 1 g(z )g(z ) 1 i j 1 z1z¯2 − = 1+¯z z −  0. " 1 ziz¯j # 1 2 1 ≥ − i,j=1 1 z¯ z  − 1 2  Therefore, the determinant of this matrix is nonnegative, i.e.,  1+ z z¯ 2 1 2 1 1+ z z¯ 1 z z¯ 1 z z¯ ≤ ⇔ | 1 2|≤| − 1 2| 1 2 − which, in turn, is equivalent to

Re(z z ) 0. (5.34) 1 2 ≤ Actually, it follows that the points z1 and z2 in D are, respectively, a pole and a zero of some function f 1 of the form (5.32) if and only if condition (5.34) is satisfied. Although we can use∈ Theorem C 5.1 to get a linear fractional representation of g, we will proceed differently. Due to the first condition in (5.33), the function g(z) z 1 zz¯ s(z)= − 1 − 1 (5.35) 1 g(z)¯z · z z − 1 − 1 belong to by the Schwarz-Pick lemma. We can write the latter equality equivalently as S z z1 − s(z)+ z1 1 zz¯1 g(z)= − z z . (5.36) 1+ − 1 z¯ s(z) 1 zz¯ 1 − 1 Combining the second condition in (5.33) with (5.35) gives z + z 1 z z¯ s(z )= A := 1 2 − 2 1 . (5.37) 2 −1+ z z¯ · z z 2 1 2 − 1 By virtue of (5.36), z z − 2 (z)+ A 1 zz¯2 E s(z)= − z z , (5.38) 1+ − 2 A¯ (z) 1 zz¯ E − 2 where A is given in (5.37). We now substitute (5.38) into (5.36): z z − 2 (z)+ A z z1 1 zz¯2 E − − z z + z1 1 zz¯1 · 2 ¯ − 1+ − A (z) 1 zz¯2 E N(z) g(z)= −z z = . (5.39) − 2 (z)+ A D(z) z z1 1 zz¯2 E 1+ − z¯1 − z z 1 zz¯1 · 1+ − 2 A¯ (z) − 1 zz¯ E − 2 26 YAEL GILBOA Long but straightforward calculations show that 2(z z )(1 z 2) (z z )¯z (z) z (1 zz¯ ) z g(z)= − 1 −| 1| − 2 2E 2 − 2 − − D(z)(1 zz¯ ) · (¯z z¯ )(1 + z z¯ ) − (z z )(1 + z z¯ ) − 2 2 − 1 1 2 2 − 1 2 1 and 2(z z )(1 z 2) (zz¯ + z z¯ ) (z) z + zz z¯ z + g(z)= − 2 −| 1| 1 1 2 E + 1 2 1 . − D(z)(1 zz¯ ) · (¯z z¯ )(1 + z z¯ ) (z z )(1 + z z¯ ) − 1 2 − 1 1 2 2 − 1 2 1 We summarize these calculations for z g(z). The result for z + g(z) is derived similarly. −

z z1 − s(z)+ z1 1 zz¯1 z g(z)= z − z z − − 1+ − 1 z¯ s(z) 1 zz¯ 1 − 1 (5.40) z z1 z z1 z + zz1 − s(z) − s(z) z1 1 zz¯1 − 1 zz¯1 − = − z z − 1+ − 1 z¯ s(z) 1 zz¯ 1 − 1 Given that the denominator is the same for both z + g(z) and z g(z), we analyze only the numerator and ﬁnd it simpliﬁes to: −

z z z z (z z )(1 s(z))=(z z )(1 + − 2 A¯ (z) − 2 (z) A) − 1 − − 1 1 zz¯ E − 1 zz¯ E − − 2 − 2 z z2 (5.41) =(z z )( − (A¯ 1) (z)+1 A). − 1 1 zz¯ − E − − 2

Recall A from (5.37) and see that z + z 1 z z¯ 2z (1 z 2) 1 A =1+ 1 2 − 2 1 = 2 −| 1| . − 1+ z z¯ · z z (1+z ¯ z )(z z ) 2 1 2 − 1 1 2 2 − 1 Substituting this and 2¯z (1 z 2) A¯ 1= 2 −| 1| − −(1 + z z¯ )(¯z z¯ ) 1 2 2 − 1 into (5.41), adding back the denominator and simplifying, we have 2(z z )(1 z 2) (z z )¯z (z) z (1 zz¯ ) z g(z)= − 1 −| 1| − 2 2E 2 − 2 . − − D(z)(1 zz¯ ) · (¯z z¯ )(1 + z z¯ ) − (z z )(1 + z z¯ ) − 2 2 − 1 1 2 2 − 1 2 1 Using the two equalities for z + g(z) and z g(z) above, we substitute into (5.32) and get: − (z z )(1 zz¯ ) f(z)= − 2 − 1 Φ(z), (5.42) (z z )(1 zz¯ ) · − 1 − 2 POLES AND ZEROS OF GENERALIZED CARATHEODORY´ CLASS FUNCTIONS 27 where (zz¯ + z z¯ ) (z) z + zz z¯ 1 1 2 E + 1 2 1 (¯z z¯ )(1 + z z¯ ) (z z )(1 + z z¯ ) Φ(z)= 2 − 1 1 2 2 − 1 2 1 . (z z )¯z (z) z (1 zz¯ ) − 2 2E 2 − 2 (¯z z¯ )(1 + z z¯ ) − (z z )(1 + z z¯ ) 2 − 1 1 2 2 − 1 2 1 Let us show that Φ(z) has a nonnegative real part, and thus is in the class, whenever condition (5.34) is satisfied. C Multiplying both numerator and denominator by (¯z z¯ )(1 + z z¯ ), we have 2 − 1 1 2 (z + zz z¯ )(¯z z¯ )(1 + z z¯ ) (zz¯ + z z¯ ) (z)+ 1 2 1 2 − 1 1 2 1 1 2 E (z z )(1 + z z¯ ) Φ(z)= 2 − 1 2 1 . z (1 zz¯ )(¯z z¯ )(1 + z z¯ ) (z z )¯z (z) 2 − 2 2 − 1 1 2 − 2 2E − (z z )(1 + z z¯ ) 2 − 1 2 1 (¯z z¯ )(1 + z z¯ ) Introducing the unimodular number Γ = 2 − 1 1 2 allows us to write the last (z2 z1)(1 + z2z¯1) formula as − (zz¯ + z z¯ ) (z)+(z + zz z¯ )Γ Φ(z)= 1 1 2 E 1 2 1 . (z z )¯z (z) z (1 zz¯ )Γ − 2 2E − 2 − 2 Let us now demonstrate that Φ(z)+ Φ(z) 0. ≥ Φ(z)+ Φ(z)= (z) 2(z z¯ +z ¯ z )( z 2 z 2) |E | 1 2 1 2 | | −| 2| 2 + (z)Γ¯z2(z1z¯2 +z ¯1z2)( z 1) E | | − (5.43) + (z)Γz (z z¯ +z ¯ z )( z 2 1) E 2 1 2 1 2 | | − + Γ 2(z z¯ +z ¯ z )( z 2 z 2 1). | | 1 2 1 2 | | | 2| − Keeping in mind that Γ is unimodular, this simplifies to: Φ(z)+ Φ(z)=(z z¯ +z ¯ z )[ (z) 2( z 2 z 2) 1 2 1 2 |E | | | −| 2| + (z)Γ¯z ( z 2 1) + (z)Γz ( z 2 1) E 2 | | − E 2 | | − 2 2 +( z z2 1)] | | | | − (5.44) = (z z¯ +z ¯ z )[ (z) 2( z 2 z 2) − 1 2 1 2 |E | | 2| −| | + (z)Γ¯z (1 z 2)+ (z)Γz (1 z 2) E 2 −| | E 2 −| | + (1 z 2 z 2)] −| | | 2| Factoring out 1 z 2, completing the square and factoring again, we have −| | Φ(z)+ Φ(z)= (z z¯ +z ¯ z )[(1 z 2)( (z)+Γz 2)+(1 z 2)(1 (z) 2)] (5.45) − 1 2 1 2 −| | |E 2| −| 2| − |E | Given that each of these pieces is nonnegative, except for (z1z¯2 +¯z1z2), which is necessarily nonpositive by (5.34), then (5.45) is nonnegative. The result of this example is the direction for future research. 28 YAEL GILBOA 6. Future Research and Acknowledgments The objective of this paper was to find something similar to the Krein-Langer representation for f Cκ. Our result was a representation of generalized Carathéodory class functions in terms∈ of Carathéodory class functions, as shown in (5.28). However, there is still work left to be done in reducing the equation to a form that can account for both the zeros and poles of the function simultaneously. The formula (5.45) shows that this is possible for the example given the constraint of (5.34). We hope that this result can be obtained for n poles and n zeros, including boundary poles and boundary zeros. “If I have seen further it is by standing on the shoulders of giants.” Sir Isaac Newton I would like to thank, first and foremost, Professor Vladimir Bolotnikov, without whom this thesis could not have been accomplished. I am eternally grateful for his endless patience and guidance, and for fortuitously taking his Complex Analysis class, which led to this thesis. Even in the darkest of times, when the end goal seemed impossibly far, Professor Bolotnikov kept me on track and helped make the overall experience an enjoyable one. Thank you to Professors Julie Agnew, Leiba Rodman and Ilya Spitkovsky for sitting on my thesis committee. Thank you to the Charles Center, as well as the Departments of Mathematics and Finance at the College of William and Mary for providing me with the opportunity to do undergraduate research. For the inspiration to pursue a thesis in the Mathematics, I must thank my mother. For support through stressful times, I thank my understanding family and friends, in particular Will Jordan-Cooley, who kept me sane throughout this process. POLES AND ZEROS OF GENERALIZED CARATHEODORY´ CLASS FUNCTIONS 29 References

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