Arov--Krein entropy functionals and indefinite interpolation problems
I. Roitberg, A.L. Sakhnovich

TL;DR
This paper extends the Arov-Krein entropy functional to generalized Nevanlinna functions, providing a representation for solutions to indefinite interpolation problems and exploring applications to indefinite Carathéodory problems and Szegő limit formulas.
Contribution
It introduces a generalized entropy functional for Nevanlinna functions and links it to indefinite interpolation problems, advancing the theory of nonclassical interpolation.
Findings
Representation of entropy functionals on indefinite interpolation solutions
Application to indefinite Carathéodory problem
Analysis of Szegő limit formula in nonclassical case
Abstract
We generalize the notion of the Arov-Krein entropy functional for the case of generalized Nevanlinna functions and obtain a representation of these functionals on solutions of indefinite interpolation problems. The case of indefinite Caratheodory problem and application to Szeg\H{o} limit formula for this nonclassical case are considered in greater detail.
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1004
Arov–Krein entropy functionals and indefinite interpolation problems
I. Roitberg and A.L. Sakhnovich
Abstract
We generalize the notion of the Arov-Krein entropy functional for the case of generalized Nevanlinna functions and obtain a representation of these functionals on solutions of indefinite interpolation problems. The case of indefinite Caratheodory problem and application to Szegő limit formula for this nonclassical case are considered in greater detail.
MSC(2010): 15B05; 28D20; 30E05; 47B35
Keywords: Arov-Krein entropy functional, generalized Nevanlinna function, indefinite Caratheodory problem, Szegő limit formula.
1 Introduction
1. In their important work [1], D.Z. Arov and M.G. Krein studied matrix-valued functions (given as linear fractional transformations), such that on the unit disk , where stands for the complex conjugate transpose of . (See also related results in [2, 26].) The class of such matrix-valued functions (matrix functions) is called Caratheodory class (or ) and admits Herglotz representation
[TABLE]
where is a nondecreasing matrix function. In particular, one can easily see that the real part
[TABLE]
The Arov–Krein entropy functionals are given by the formula:
[TABLE]
Further results on Arov-Krein and related Burg’s entropy and further references are given in [4, 5, 7, 11, 14].
Here, we consider a natural generalization of the Arov–Krein entropy functional for the case of the solutions of indefinite interpolation problems belonging to the Krein–Langer (generalized Caratheodory) class , where . More precisely, we take solutions of indefinite interpolation problems belonging to the generalized Nevanlinna class , transform them into and consider the entropy. See Definition 1.1 of the classes and and a more detailed discussion below. The problem is closely related to the Szegő limit formula in the indefinite case [27, 28] (see also interesting recent papers [8, 9]).
2. The solutions of the so called indefinite interpolation problems are often described in terms of linear fractional transformations. In this paper, we deal with the case when the matrix functions, which are obtained using these linear fractional transformations, are defined in the open upper half-plane . Moreover, these functions belong to the classes , where (see the definition of below).
Definition 1.1
The generalized Nevanlinna class is the set of meromorphic matrix functions such that the kernel \big{(}\varphi(z)-\varphi(\zeta)^{*}\big{)}\big{/}(z-\overline{\zeta}) has negative squares. That is, for any and any set the matrix \big{\{}\big{(}\varphi(z_{i})-\varphi(z_{k})^{*}\big{)}\big{/}(z_{i}-\overline{z_{k}})\big{\}}_{i,k=1}^{n} has at most negative eigenvalues and for at least one choice it has exactly negative eigenvalues.
The generalized Caratheodory class is the set of meromorphic matrix functions such that the kernel \big{(}\omega(\lambda)+\omega(\zeta)^{*}\big{)}\big{/}(1-\lambda\overline{\zeta}) has negative squares. That is, for any and any set the matrix \big{\{}\big{(}\omega(\lambda_{i})+\omega(\zeta_{k})^{*}\big{)}\big{/}(1-\lambda_{i}\overline{\zeta_{k}}\big{\}}_{i,k=1}^{n} has at most negative eigenvalues and for at least one choice it has exactly negative eigenvalues.
Here denotes the set of natural numbers and is the complex conjugate of . The classes and have been studied in a series of seminal Krein–Langer papers, for instance [16, 17] (see also the articles [6, 21] and references therein). Clearly, .
Remark 1.2
Sometimes, we use the notations and instead of and , respectively, to stress the order of the matrix functions in the classes.
We will use the “operator identities” formalism of solving indefinite interpolation problems from [27] (scalar function case) and [22, 23, 28] (matrix function case), which is based on the approaches to sign-definite interpolation problems described in [20, 33]. This formalism as well as the representation of the generalized Nevanlinna functions is discussed in Section “Preliminaries” (Section 2). In Section 3, we introduce entropy functional for the generalized Nevanlinna functions and obtain a representation of this functional on the solutions of indefinite interpolation problems. In Section 4, we consider in detail the case of block Toeplitz matrices and indefinite Caratheodory problem. Section 5 is dedicated to the application of the results from Section 4 in the proof of indefinite Szegő limit formula.
In the paper, denotes the set of natural numbers, denotes the set of nonnegative integers, denotes the real axis, is the unit disk (i.e., ), stands for the unit circle (i.e., ), stands for the complex plane, and () stands for the open upper (lower) half-plane. For , is the characteristic function of , that is, for and for . The spectrum of a bounded operator or a square matrix is denoted by and stands for the identity matrix.
The set of linear bounded operators acting from the Hilbert space into Hilbert space is denoted by and the notation we simplify into . The set of Hermitian operators (or matrices) from , such that the spectrum of contains precisely (counting multiplicities) negative eigenvalues (), is denoted by . We use also the notation for the mentioned above .
2 Preliminaries
2.1 Representation of
The matrix function admits representation [6] (see also [21]):
[TABLE]
where are real numbers, are distinct numbers;
(i) the real line is a union of the sets , such that are bounded open intervals having disjoint closures, is there complement, and ;
[TABLE]
(recall the definition of the characteristic function from the introduction),
[TABLE]
(ii) is a nondecreasing (on each of the intervals , , where , and ) matrix function such that the following integral converges:
[TABLE]
(iii) for each , the matrix function is a matrix polynomial of degree at most having self-adjoint matrix coefficients, such that the coefficient in the term of maximal degree in is nonnegative (), and the equalities hold;
(iv) for each , the matrix function is a matrix polynomial such that .
Remark 2.1
It is easy to see that
[TABLE]
Notation 2.2
The degree of is denoted by .
Remark 2.3
For , without loss of generality see [16, Theorem 3.1] we require that
[TABLE]
2.2 Indefinite interpolation problem
Let the operator (or matrix) be given. In the following text we assume that for some and the operator identity
[TABLE]
holds. Fixed operators and determine a class of so called structured operators satisfying (2.8) (e.g., Toeplitz or Loewner matrices, operators with difference kernels and so on). We assume also that is invertible and
[TABLE]
1. We introduce the operators and in terms of the representation (2.1) of . For this purpose we need some preparations. Let and be fixed and the matrix function and be given. Then we set
[TABLE]
Definition 2.4
The Krein–Langer data corresponding to the representation (2.1) is the set
[TABLE]
Clearly the data corresponding to is not defined uniquely although the arbitrariness is not so great (see Remark 2.6). In particular, the points and the functions are fixed. The following operators and operator functions are generated by the data using (2.10)–(2.13).
Definition 2.5
When F_{j}(z)=R_{j}\big{(}1\big{/}(z-\alpha_{j})\big{)} and , we denote the corresponding and by and , respectively; when F_{j}(z)=M_{j}\big{(}1\big{/}(z-\beta_{j})\big{)} and , we denote the corresponding and by and , respectively, and when F_{j}(z)=M_{j}\big{(}1\big{/}(\overline{z}-\beta_{j})\big{)}^{*} and we denote the corresponding by . When and , we denote the corresponding by . When , we denote the corresponding and by and , respectively. Finally, setting , where is an additional parameter, we put
[TABLE]
The matrix functions are introduced by the similar to (2.10) formulas
[TABLE]
for , and
[TABLE]
Let , and the representation (2.1) of be given. Then, taking into account Definition 2.5, we introduce the operators
[TABLE]
Remark 2.6
According to [22, Section 3] the operators and are well-defined and the integrals in (2.17) and (2.18) weakly converge under conditions (2.27) and
[TABLE]
Moreover, Theorems 3.4 and 3.5 in [22] state that and do not depend on the choice of the domains , that , where , and the operator identity
[TABLE]
holds.
2. The transfer matrix function in Lev Sakhnovich form [29, 31, 33] is introduced by the equality
[TABLE]
where , and satisfy (2.8), is the identity operator and is the identity matrix. We will also need the matrix function
[TABLE]
where and are blocks of . By virtue of the properties of the transfer matrix functions (see, e.g. [29, (1.84)]) we have a useful equality
[TABLE]
The linear fractional transformations, which we are interested in, are given by the formula
[TABLE]
where the matrix functions and are meromorphic in and satisfy the inequalities
[TABLE]
except at isolated points. One says that such pairs are nonsingular pairs with the property-. Given a so called frame , let us introduce .
Notation 2.7
The set of matrix functions of the form (2.24), where are nonsingular pairs with the property- which satisfy (2.25), is denoted by .
In this paper, we consider the simpler interpolation cases, where the following conditions are valid:
[TABLE]
3. The function given in is determined (further in the text) in by the relation
[TABLE]
Now, we introduce in by the equality
[TABLE]
Notation 2.8
We denote by the class of matrix functions which are analytic in excluding, possibly, isolated points and may have only removable singularities in the points such that .
The next interpolation theorem directly follows from [22, Theorems 4.4, 5.1].
Theorem 2.9
*Let the operators , , and satisfy the operator identity (2.8). Assume that , that conditions (2.9) and (2.27) are fulfilled and that is given by (2.22). Then the following two statements are valid.
If in , see Notation 2.8 and the function is analytic at every such that , then , (2.19) is valid and the equalities*
[TABLE]
*where and are given by (2.17) and (2.18), respectively, are satisfied.
Conversely, if some matrix function belongs and relations (2.19) and (2.30) are fulfilled, then , and is analytic at every such that .*
Clearly, one can consider the linear fractional transformation (2.24) in both half-planes and as is done in [22, 27] and is explained for the case below.
Definition 2.10
Let the conditions of Theorem 2.9 hold. Then, the matrix functions considered in Theorem 2.9 are called the solutions of the interpolation problem (2.30).
In the scalar case , the pairs look somewhat simpler (see the remark below and [27]).
Remark 2.11
Assuming in this remark, we note that without loss of generality one may consider linear fractional transformations (2.24) generated by the pairs and completed with the pair and . In other words, we have
[TABLE]
where
[TABLE]
We set . Formula (2.23) may be rewritten in the form
[TABLE]
When , formula (2.33) takes the form
[TABLE]
Recall that
[TABLE]
In view of (2.34) and (2.35), the functions given in (2.31) for have the same form in , that is
[TABLE]
Instead of the condition in Theorem 2.9, we will use the condition when in the next theorem.
Theorem 2.12
*Let and let the operators , , and satisfy the operator identity (2.8). Assume that and that conditions (2.9) and (2.27) are fulfilled. Then the following two statements are valid.
If and when \varphi=\mathrm{i}a(z)\big{/}c(z)) for all such that , then and the equalities (2.19) and (2.30) are satisfied.
Conversely, if some function belongs and relations (2.19) and (2.30) are fulfilled, then .*
3 Entropy
There is a simple one to one mapping between the matrix functions and matrix functions (see, e.g., [16] or [23, p. 344]):
[TABLE]
Correspondingly, the representation (2.1) of is equivalent to the following representation of :
[TABLE]
where is a nondecreasing on the intervals , for and matrix function, and the following integral converges:
[TABLE]
(i.e., or ); for , where the set is taken from the representation (2.1) of ;
is a characteristic function and for the set is an open interval on containing ; ;
[TABLE]
for we have
[TABLE]
and for we have
[TABLE]
According to [23, formula (2.10)] we have
[TABLE]
where and take the values and , respectively, as well as , and , . We have also (see [23, formula (2.9)]) a useful relation
[TABLE]
Definition 3.1
The entropy functionals on the matrix function are given by the formula
[TABLE]
where is the matrix function from the Krein–Langer representation (3.2) of and and are constructed from and , respectively, using (3.1).
Note that the right-hand sides of (1.3) and (3.9) formally coincide although the requirements on in (1.3) and (3.9) differ.
Consider representation (3.2) in greater detail. Fix some arbitrary closed interval belonging to one of the intervals , , where , or , and note that the matrix function is bounded on . Hence, we write down as a sum of two functions generated by the representation (3.2):
[TABLE]
It is easy to see that . Taking into account Smirnov’s and Nevanlinna’s theorems, we derive that the entries of belong to the Hardy class for each , and the non-tangential limits exist and are finite almost everywhere. (One could use also theorem for analytic functions with positive real part from [15, p. 58].) In view of the representation (3.2) of and equalities (1.2), (3.4) and (3.8), it is easy to see that the function has the following property:
[TABLE]
Here we used the relations (3.3) and (3.7) as well. Hence, according to (3.10) the following equality holds for the non-tangential limits:
[TABLE]
Therefore, we denote the limits in (3.12) as \Re\big{(}\omega(\mathrm{e}^{\mathrm{i}t})\big{)} and \Re\big{(}\widetilde{\omega}(\mathrm{e}^{\mathrm{i}t})\big{)}, and (3.12) takes the form
[TABLE]
Now, using the definition of in (3.10) and Fatou’s theorem (see, e.g., [19, p. 39]) we have \Re\big{(}\widetilde{\omega}(\mathrm{e}^{\mathrm{i}t})\big{)}=\mu^{\prime}(t) for . Thus, almost everywhere on equality (3.13) yields
[TABLE]
In view of (3.14), we rewrite (3.9) in the form
[TABLE]
Next, we turn to the linear fractional transformations (2.24). According to (2.22), (2.24), and (2.25) we have
[TABLE]
Recalling (2.23), we see that
[TABLE]
Moreover, (3.1) implies that
[TABLE]
and we set, correspondingly,
[TABLE]
Using (3.16), (3.17) and (3.19), we rewrite (3.15) in the form
[TABLE]
Let us study first the case of the following nonsingular pairs with the property-:
[TABLE]
Each nonsingular pair with the property-, such that , may be substituted by the equivalent pair (generating the same ) of the form (3.21). Using (3.21), we rewrite (3.20) in the form
[TABLE]
where is defined in (3.19).
Remark 3.2
Formula (3.22) is a generalization for the sign-indefinite case of the formula [2, (11)] see the corresponding Theorems 2 and 3 in [2].
The general case of the pairs is dealt with in Remark 4.8.
Remark 3.3
Our next considerations are similar to the corresponding considerations in the proof of the Szegő limit theorem indefinite case, see [28, pp. 480-482]. In particular, we use the fact that the entries of , where and is taken from the representation (3.2) of , belong to some Hardy class . This fact follows from the expressions [23, (2.8)] for and from the convergence inequality (3.3). As a result, for the entries of we have for some the inequalities:
[TABLE]
Further we assume that
[TABLE]
and so and are matrices.
Notation 3.4
Let us introduce the notation
[TABLE]
where is given in (3.18).
The number of different eigenvalues of in we denote by .
Proposition 3.5
Let , , and (2.8) hold. Assume that (2.27) is valid and . Then, the number of different zeros of in is finite. Moreover, we have
[TABLE]
P r o o f
. Similar to [30] and [28, (1.17)] we introduce the matrix function
[TABLE]
where
[TABLE]
From (2.8) and (3.27) we derive
[TABLE]
In this proof we will consider and for . Assume first that
[TABLE]
In this case, is -dissipative and does not have real eigenvalues. The first fact follows from (3.30) and the second fact is proved by contradiction. Indeed, assuming that
[TABLE]
and multiplying both parts of (3.29) by from the left and by from the right, we obtain
[TABLE]
Recall that is sign-definite, and so equality (3.32) yields . Hence, taking into account (3.27) and (3.31), we derive . Since (in view of (2.27)), does not have real eigenvalues, we arrive to a contradiction.
We showed that is -dissipative and does not have real eigenvalues. Recall also that . Thus, has eigenvalues in counting multiplicities. In the same way as in [28, p. 477] it follows that has no more than zeros (counting multiplicities) in for all . We note that may now depend on and means .
According to (3.27) we have the equality
[TABLE]
which we rewrite in the form
[TABLE]
where
[TABLE]
On the other hand formula (2.22) implies that
[TABLE]
Finally, we rewrite (3.34) in the form
[TABLE]
Recall that the number of different zeros of in equals and that has no more than zeros (counting multiplicities) in . Hence, the same holds for and , respectively. Taking into account relations (3.35), (3.37) and , we see that G\big{(}1\big{/}\overline{z}\big{)}^{*} is holomorphic and does not have zeros in excluding, possibly zeros of and . Hence, (3.36) yields that the number of different zeros of
[TABLE]
in is less or equal . Clearly, the same is valid, if we switch from to , where is given by (3.18), and consider zeros of the function (see (3.25)) in . That is, (3.26) is proved.
Notation 3.6
Different zeros of are denoted by and their multiplicities are denoted by , respectively.
It follows from (2.22) that is analytic in . Moreover, since the entries of belong to , formula (2.22) yields that belongs to some Hardy class as well. Hence, using Proposition 3.5 and Notation 3.6 we write down in the form
[TABLE]
where is the Blaschke product and belongs to for some and does not have zeros in .
Theorem 3.7
Let (3.21) and (3.24) hold, assume that the matrices , , and satisfy the matrix identity (2.8), that satisfies (2.27), that , that and that .
Then, the entropy functional on where is given by (2.22) satisfies the equality
[TABLE]
if only the terms on the right-hand side above are finite. Moreover, if we set for and require additionally that
[TABLE]
then the matrix functions are solutions of indefinite interpolations problems (2.30) and formula (3.39) provides the values of entropy functionals for these solutions.
P r o o f
. Together with the matrix function we consider the matrix function \big{(}c(z)\psi(z)+\mathrm{i}d(z)\big{)}^{-1}. It is immediate from (2.22) and (2.24) that
[TABLE]
Relation (2.23) implies that , and we rewrite (3.41) in the form
[TABLE]
In particular, we have
[TABLE]
According to (3.1), (3.25), (3.38) and (3.43) we have
[TABLE]
Using (2.22), (3.23) and (3.44) we obtain
[TABLE]
for some . Formula (3.45) means that belongs some Hardy class, and we recall that belongs some Hardy class as well. Thus, is an outer function. It is immediate from the parameter representation of the outer function (see, e.g. [15, p. 76]) that
[TABLE]
Next, using (3.25) and (3.38) we derive
[TABLE]
where is given in (3.19). We note that according to (3.38) the equality
[TABLE]
is valid. In view of (3.46)–(3.48), we rewrite (3.22) in the form
[TABLE]
Now, (3.25) and (3.49) imply (3.39).
Taking into account [22, Theorem 5.2] (for the case ), we see that the conditions of Theorems 2.9 and 2.12 are fulfilled if (3.40) holds. In other words, if (3.40) holds, the matrix functions are solutions of indefinite interpolations problems (2.30).
Remark 3.8
In the proof of Proposition 3.5, we studied starting from (3.30) the case of . We could substitute the inequality in (3.30) with , in which case would be -dissipative. Quite similar to the considerations after (3.30) one shows that has no more than zeros counting multiplicities in . Here in and is introduced via (3.28) in . Clearly, relations (3.34)–(3.37) hold in .
Remark 3.9
If one wants to consider entropy for all the solutions of indefinite interpolation problem one may easily swithch from the matrix functions to contractive matrix functions . See Remarm 4.8, Definition 4.9 and formula (4.50) as well as the considerations after (4.50).
4 Entropy functionals and indefinite
Caratheodory problem
Consider the case of block Toeplitz matrices, that is, the case
[TABLE]
where are blocks. Block Toeplitz matrices are unique solutions of the matrix identities (see [24, 25, 28]):
[TABLE]
where
[TABLE]
For related results and discussions on the operator identities or operators and matrices with the displacement structure see, for instance, [12, 13, 29, 31, 32]. The next proposition is immediate from (4.1)–(4.15).
Proposition 4.1
The matrices , , and given by (4.1), (4.6) and (4.15) satisfy conditions on these matrices from Theorems 2.9 and 3.7, and the corresponding space , where and act, is finite dimensional. More precisely, .
Let us add index in the notation and write instead of when we consider matrices. We say that if all the reductions starting from some belong to .
Definition 4.2
Matrices are called extensions of the matrix .
The following result may be derived as a special case of Theorem 2.9 and presents a reformulation of [28, Theorem 4.1].
Theorem 4.3
Let the matrices , and by given by (4.6) and (4.15), let be given by the equalities in (4.1), and let the relations and hold. Assume that the matrix functions are given by (2.24), where is given by (2.22) and the pairs are nonsingular pairs with the property- satisfying inequality
[TABLE]
Then the functions admit Taylor expansions with the same first Taylor coefficients determined by
[TABLE]
Moreover, further Taylor coefficients of the functions generate extensions of belonging to the classes
[TABLE]
, and all the extensions of belonging to the classes are generated in this way.
While reformulating [28, Theorem 4.1] as Theorem 4.3 we took into account that the coefficients (see [28, (1.6)]) of the linear fractional transformation of the form [28, (1.8)] and given by (2.22) are connected by the relation:
[TABLE]
Formula (4.20) implies the equalities , where the matrix functions are given by [28, (1.8)], the matrix functions are given by (2.24), and there is a simple one to one correspondence between the matrix functions and nonsingular pairs with the property-.
Next, let us consider condition (4.17) in greater detail. We start with a simple auxiliary lemma.
Lemma 4.4
Let and be matrices, let all four matrices have rank and assume that the equalities
[TABLE]
hold. Then, the inequalities
[TABLE]
are equivalent.
P r o o f
. Let us show that the first inequality in (4.22) yields the second (clearly, the fact that the second inequality yields the first is proved in the same way). Indeed, if there is such that . Hence, in view of , there is such that , and therefore implies that , which contradicts .
Lemma 4.5
Let the matrices , and by given by (4.6) and (4.15), let be given by the equalities in (4.1), and let the relations and hold. Assume that the pair is given by (3.21). Then, condition (4.17) is equivalent to the condition
[TABLE]
P r o o f
. According to [28, p. 452] we have
[TABLE]
Setting
[TABLE]
and using (2.22) and (4.24), we see that
[TABLE]
Moreover, formulas (2.22), (2.23) and (4.26) imply the equality
[TABLE]
where, in view of (2.22) and (4.24), we have
[TABLE]
We will need the equalities
[TABLE]
In order to show that rank , we partition the following matrices into the blocks:
[TABLE]
It easily follows from (4.2) and is immediate from [25, (12)] that
[TABLE]
Thus, if rank , we have for some , and formulas (4.30)–(4.32) yield
[TABLE]
which contradicts the invertibility of . Hence, indeed, rank .
Using the equality rank , one can show that there is an extension such that (see, e.g., [25, Lemma 8]). Now, relations (4.28) and [28, (2.16)] yield the equality
[TABLE]
and (4.29) is proved.
Finally we set
[TABLE]
Taking into account (4.27) and (4.29), we see that and satisfy conditions of Lemma 4.4. It follows that the inequalities
[TABLE]
and (4.23) are equivalent.
In order to derive the entropy formula and include the important case one could remove singularities of \det\big{(}c(z(\lambda))\psi(z(\lambda))+\mathrm{i}d(z(\lambda))\big{)} inside in a simpler way than in (3.25) and introduce as the product:
[TABLE]
We note that in the case (4.1)–(4.15) of Toeplitz matrices (and in view of (3.18), where ) the following equality holds:
[TABLE]
Hence, it will be even more convenient to consider of the form
[TABLE]
instead of given by (4.35). Now, from (2.22), (3.18), (4.24) and (4.37) we derive
[TABLE]
We introduce as the Blashke product in the representation of , that is, in terms of zeros of counting multiplicities is given by the formula
[TABLE]
In a similar to the proof of Theorem 3.7 way one can show that
[TABLE]
where is an outer function.
Theorem 4.6
Let the matrices , and be given by (4.6) and (4.15), let be given by the equalities in (4.1), and let the relations and hold. Assume that the pair has the form as in (3.21) and that (4.34) is valid. Set in (3.1), (3.18) and (3.19). Then, defined by (2.24) is a solution of the interpolation problem (2.30) as well as the indefinite Caratheodory problem described in Theorem 4.3 and its entropy is given by the formula:
[TABLE]
where and are given by (4.37) and (4.39), respectively.
P r o o f
. It is easy to see that the conditions of Theorem 4.3 are fulfilled. According to Lemma 4.5, inequality (4.34) is equivalent to (4.23), which in our case coincides with (3.40). Thus, satisfies the conditions of Theorem 3.7 as well. Therefore, is a solution of the interpolation problem (2.30) and of the indefinite Caratheodory problem described in Theorem 4.3.
Taking into account that \big{|}\widetilde{B}\big{(}\mathrm{e}^{\mathrm{i}\theta}\big{)}\big{|}=1, that and that equalities (4.37) and (4.40) hold, in the same way as in the proof of (3.39), we rewrite (3.22) in the form (4.41).
Formulas (4.25) and (4.38) yield
[TABLE]
Remark 4.7
Relations (4.34) and (4.42) imply that is well-defined at and moreover . Since and , one can consider entropy at .
Remark 4.8
Sometimes, it is more convenient to use contractive matrix functions instead of the nonsingular pairs with the property-. For this purpose, one introduces matrix with the following property:
[TABLE]
We set:
[TABLE]
From (4.43) and (4.44), it is easy to see that
[TABLE]
and so is invertible and is contractive when the pair is nonsingular with the property-. Moreover, relations (2.24), (4.43) and (4.44) imply that
[TABLE]
where
[TABLE]
Clearly, inequality (2.25) is equivalent to the inequality
[TABLE]
and so see Notation 2.7 coincides with the set of functions , where are contractive in and (4.48) holds.
Definition 4.9
According to [2, (1)], the entropy of the matrix function which is contractive in is given by the formula
[TABLE]
Taking into account (4.43), (4.44), (4.47) and (4.49) we rewrite (3.20) in the form
[TABLE]
where is given by (3.19).
In the case of indefinite Caratheodory problem, we (similar to (4.37) and (4.39)) set
[TABLE]
where are zeros of counting multiplicities. Next, we factorise : and in the same way as (3.44) we obtain the equality
[TABLE]
Using (3.23) and (4.53), one can see that is again an outer function. Hence, taking into account (4.50), we have a somewhat stronger version of Theorem 4.6.
Theorem 4.10
Let the matrices , and be given by (4.6) and (4.15), let be given by the equalities in (4.1), and let the relations and hold. Set in (3.1), (3.18) and (3.19). Assume that the matrix functions are contractive in and satisfy the inequality
[TABLE]
Then, the set of matrix functions of the form (4.46) describes via Taylor coefficients of as in Theorem 4.3 all the solutions of the indefinite Caratheodory problem. The entropy functional on these matrix functions is given by the formula:
[TABLE]
5 Nonclassical Szegő limit formula
In order to introduce the entropy functionals, we transformed functions into the functions belonging to the class . It is easy to see that the relation yields . In precisely the same way as before, one can deal with the entropy generated by the transformation of into . In particular, relation (3.15) takes the form
[TABLE]
It is convenient to consider instead of in the case of Caratheodory problem. Indeed setting in (3.18) we obtain
[TABLE]
that is coincides with the function on the left-hand sides of important formulas (4.18) and (4.19). Using (5.1) and the same arguments as in the proof of Theorem 4.6, one rewrites (4.41) in the form
[TABLE]
Now, let us fix , and . In the same way as generates block Toeplitz matrices using Taylor coefficients from (4.19), the matrix function generates block Toeplitz matrices . Since , we have for all . The notations, which we introduce using (e.g., and ) will obtain an accent ”breve” if we substitute with (and we will write and in that case). Introduce also the notations:
[TABLE]
When , the famous first Szegő limit formula is valid:
[TABLE]
On the other hand, formulas (5.1) and (5.2) imply that
[TABLE]
According to (5.5) and (5.7) we have
[TABLE]
The following important correspondences exist between the matrices generated by and (see [28, (3.5), (3.6), (3.40)])
[TABLE]
where the matrix does not depend on and () is the right lower block of (of ):
[TABLE]
From (4.42) and [28, (4.39)] it follows that
[TABLE]
Moreover, we have (see, e.g., [28, (4.31)]):
[TABLE]
[TABLE]
Using the entropy formula (5.3) (at ) and taking into account (4.39), we rewrite (5.12) in the form
[TABLE]
From the formula above, in view of (5.1) and (5.2) we derive the nonclassical Szegő limit formula for matrices generated by :
[TABLE]
where are the poles of counting multiplicities. For further details see [28, Section 4]. On the asymptotics of determinants in some other important nonclassical cases see, for instance, [3, 10].
Acknowledgments. This research was supported by the Austrian Science Fund (FWF) under Grant No. P29177.
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