Some Propositions on Generalized Nevanlinna Functions of the Class

Abstract

Some propositions on the generalized Nevanlinna functions are derived. We indicate mainly that (1) the negative inertia index of a Hermitian generalized Loewner matrix generated by a generalized Nevanlinna function in the class does not exceed . This leads to an equivalent definition of a generalized Nevanlinna function; (2) if a generalized Nevanlinna function in the class has a uniform asymptotic expansion at a real point or at infinity, then the negative inertia index of the Hankel matrix constructed with the partial coefficients of that asymptotic expansion does not exceed . Also, an explicit formula for the negative index of a real rational function is given by using relations among Loewner, Bézout, and Hankel matrices. These results will provide first tools for the solution of the indefinite truncated moment problems together with the multiple Nevanlinna-Pick interpolation problems in the class based on the so-called Hankel vector approach.

1. Introduction

Let , , and be the complex plane, the real axis in the complex plane, and the set of nonnegative integers, respectively. Denote by and the open upper half complex plane and the set of positive integers, respectively. For a meromorphic function of one complex variable, its domain of holomorphy of is written as . Denote by the number of negative eigenvalues of a Hermitian matrix .

In 1977, Krein and Langer [1, Section 1] introduced the function class of all generalized Nevanlinna functions with negative index .

Recall that a meromorphic function in is called a generalized Nevanlinna function with negative index , if it satisfies the symmetric condition: for all , and, for each choice of , distinct points , the Hermitian matrix has at most negative eigenvalues (counted with multiplicities) and for at least one such choice exactly negative eigenvalues; that is,

In particular, for , the class is actually the set of all Nevanlinna functions which are holomorphic in such that and for all . It is well known that (see, e.g., [2, 3]) each admits an integral representation of the form: in which , , and is a positive measure on satisfying

By definition, we check easily that (see, e.g., [1, 4]) and if and only if , and also, if and only if . Moreover, if and , then , where .

For , the symbol stands for the sector in with the vertex : The notation means that the function has a nontangential limit at ; that is, tends to as tends to in each sector . Also, the notation stands for the nontangential limit of as in each sector .

Now, we introduce the notions of generalized zeros and generalized poles of nonpositive type of a generalized Nevanlinna function (see [5–8] for details).

A point is called a generalized zero of nonpositive type of multiplicity of if Similarly, the point is called a generalized zero of nonpositive type of multiplicity of if A point is called a generalized pole of nonpositive type of multiplicity of if Similarly, the point is called a generalized pole of nonpositive type of multiplicity of if

Remark that the point is a generalized zero of nonpositive type of multiplicity of if and only if 0 is a generalized zero of nonpositive type of multiplicity of and that the point is a generalized pole of nonpositive type of multiplicity of if and only if the point is a generalized zero of nonpositive type of multiplicity of (see [8] for the proof of sufficiency).

In 1981, Krein and Langer [7] showed that the total multiplicity of poles (zeros, respectively) in and generalized poles (generalized zeros, respectively) of nonpositive type in of is equal to , which implies that each has finite number of zeros and poles in . As a corollary of this result, we infer ([9, Proposition 3.2]) that if and have no common poles in and common generalized poles of nonpositive type in , then .

By means of an operator representation of , in 1985, Daho and Langer [9, Proposition 2.1] derived an integral representation of as follows: where is a positive measure on satisfying , are either the poles of in or its generalized poles of nonpositive type on , counted according to their multiplicities, with , and is a given positive number such that .

In 2000, Dijksma et al. [6] gave an interesting factorization for each function . More precisely, if are all the zeros in and the generalized zeros of nonpositive type of on and are all the poles in and the generalized poles of nonpositive type of on , all counted according to their multiplicities, then where . The factorization result (11) of , together with the integral representation (3) of , gives rise to the following result. Each admits an integral representation where , , , are the same as in (11), and , , and is a positive measure on satisfying .

It should be noted that generalized Nevanlinna functions are closely related to the theory of extensions of symmetric operators in the Pontryagin spaces. The reader can refer to [1, 7, 10–13] and the references therein.

In this paper, we present some interesting propositions on generalized Nevanlinna functions in the class . Firstly, we show that the number of negative eigenvalues of each Hermitian generalized Loewner matrix generated by a function is not greater than . This leads to an equivalent definition of the generalized Nevanlinna function with negative index (see Theorem 7 below), which extends the original one to the multiple point case. Secondly, we indicate that the number of negative eigenvalues of the Hankel matrix, constructed with partial coefficients of the asymptotic expansion of a function at a real point or at infinity, does not exceed (see Theorems 14 and 15). Finally, we give an explicit formula of the negative index of real rational functions by using the relations among Loewner, Bézout, and Hankel matrices (see Theorem 21). These results will provide first tools for the solution of the indefinite truncated moment problems together with the multiple Nevanlinna-Pick interpolation problems in the class , based on the so-called Hankel vector approach, in subsequent work.

2. An Equivalent Definition of Generalized Nevanlinna Functions of the Class

In this section, we will show that the Hermitian Loewner matrix appearing in the definition of generalized Nevanlinna functions with negative index can be substituted by a certain Hermitian generalized Loewner matrix. For a more general form of generalized Loewner matrices, the reader can refer to [14, 15].

For convenience, we introduce the notion of the divided difference of a complex function of one complex variable. Let be such a function of one complex variable , let be a sequence of distinct points located in the domain of , and let , and, in the case , moreover, Such is called the th divided difference of at .

In the sequel, we give some properties of the higher divided difference of the complex functions in one complex variable. These properties are very similar to those of the higher divided difference of the real functions of one real variable. But the proofs of them need to be checked carefully.

Proposition 1. Let be an open disk in the complex plane, and let be distinct points. If is holomorphic in the closed disk , then in which is an arbitrary -permutation.

Proof. By using Cauchy’s Integral Formula, we infer that Then, for each -permutation , we have The last equation implies that (14) holds for each -permutation .

With the proof of Proposition 1, we can prove the following two statements.

Proposition 2. If is holomorphic at the point and are distinct and variable points in , then holds for an arbitrary .

Proof. Since is holomorphic at , there exists such that is holomorphic in . It follows from the proof of Proposition 1 that Then (17) follows.

Proposition 3. If is holomorphic at two distinct points and are distinct and variable points in , then holds for any .

Proof. Since is holomorphic at the points and , there exist two disks and with centers at and , respectively, such that and is holomorphic in and . By using Cauchy’s Integral Formula, where . Therefore, On the other hand, Combining (21) and (22), we obtain (19) immediately.

The following lemma plays an important role by proving the fact that the number of negative eigenvalues of the Hermitian generalized Loewner matrices generated by a function does not exceed .

Lemma 4. Let be a complex function of one complex variable, and let be distinct points in the domain of . Then in which and with being nonsingular matrices for and .

Proof . Let and , in which, for , and, for , Obviously, , , and . By Proposition 1, we check easily that that is, and . Hence we have as asserted.

In particular, if, in Lemma 4, , , and , then and thus (23) in this case can be rewritten as follows:

Suppose now that and are mutually different, with multiplicities , respectively. For each , there exist mutually different points () such that Therefore, Proposition 3 says in which , , and . Since and all the are distinct points, we have Observe that in which By use of (23), there exist nonsingular matrices such that or equivalently, The last equation together with (32) implies that

Let and let be distinct points with multiplicities , respectively. Define the generalized Loewner matrix generated by the function and such by Proposition 3 indicates that where is defined by (36).

Thus, from (2) and the fact that each eigenvalue of a square matrix is a continuous function of its entries, we immediately obtain the following result.

Theorem 5. Let , and let be distinct points with multiplicities , respectively. Then the Hermitian generalized Loewner matrix defined by (38) has at most negative eigenvalues.

Proof. Since , by (2), we have for . It follows from (39) and the continuity of eigenvalues of a square matrix that Thus we complete the proof.

Observe that, by Proposition 3, each entry of defined by (38) can be rewritten as follows: Then the Loewner matrix appearing in the definition of generalized Nevanlinna functions of the class is a special case of the generalized Loewner matrix defined by (38).

Theorem 6. Let be meromorphic in satisfying for . If
then .

Proof. It follows from (43) that, for an arbitrary and for any distinct points , , and, for some choice of and distinct points ,
Let now and let . Taking small enough such that , we can choose different points in for instead of the original multiple point such that This is possible, because of (36) and (39) together with the continuity of the eigenvalues of a square matrix for its entries. By definition, we obtain from the afore-stated results that . Thus the proof is complete.

In view of Theorems 5 and 6, we present now the announced result, that is, an equivalent definition of generalized Nevanlinna functions of the class in some sense, which extends the original one to the multiple point case.

Theorem 7. Let be a meromorphic function in such that for . Then is a generalized Nevanlinna function of the class , if and only if (41) holds for any and any distinct points with multiplicities , respectively, and, for at least one such choice, the equality in (41) holds.

3. Some Propositions on Asymptotic Expansions of of the Class

In this section, we study some propositions on asymptotic expansions of a function at and at infinity. To begin with, we introduce the notion of angular derivative of a meromorphic function defined on at a real point .

Let be meromorphic in and let . If the nontangential limits exist, then we say that has an angular derivative (of order 1) at , denoted by . Sometimes we write and as and , respectively.

More generally, for an arbitrary , we say that has the th angular derivative at a real point , denoted by , if the nontangential limits exist, and in this case .

In [16], Chen and Hu proved an interesting property on higher angular derivatives of a function in the class at a real point.

Lemma 8. Let and let have an integral representation (3), in which , , and is a positive measure on such that (4) holds. Then, for some , the angular derivatives exist and are real numbers if and only if . In that case, in which stands for Kronecker symbol, that is, when , ; when , .

It follows from Lemma 8 that if exist and are real numbers then, for , Obviously, the first term on the right hand side of the inequality (49) tends to 0 when tends nontangentially to . As for the second term therein, noting for all , in the case of , we have Since , by Lebesgue’s dominated convergence theorem, the second term on the right hand side of the inequality (49) tends to 0 as well when tends nontangentially to . Thus Therefore, the following statement is valid.

Proposition 9. Let and . If, for some , the angular derivatives exist and are real numbers, then

Next, we verify that each function in the class has a similar property to that given in Proposition 9. For this purpose, we need the following lemma.

Lemma 10. Let , be meromorphic in , let , and let . If and have the angular derivatives of order from 1 to at , then

Proof. In the case of , we have Thus the equality (53) holds for . Now we assume that the equality (53) holds for ; that is, Then, for the case of , which means that, for , the equality (53) is also true. Therefore, by the principle of mathematical induction, we conclude that the equality (53) is valid for all .

Let and have the angular derivatives of order from 1 to at . Then observe that has also the angular derivatives of order from 1 to at ; moreover, On the other hand, by Lemma 10, has then the angular derivatives of order from 1 to at real . As tends nontangentially to real in (53), we have at , which implies in turn that and consequently where , .

In view of the facts (57) and (60), we can prove the following result.

Theorem 11. Let and let have an integral representation of the form (10). Then for some , the angular derivatives exist and are real numbers if and only if In that case,

Proof. Observe that the integrand in (10) can be rewritten as , so the integral representation (10) of can be rewritten as where with , and , and are as in (10). Let now Then , or equivalently, , where , . Since, for real and , from (57) and (60), we deduce that the angular derivatives exist and are real numbers if and only if the angular derivatives exist and are real numbers. Thus, by Lemma 8, the angular derivatives exist and are real numbers if and only if the inequality (61) holds.
From the relation , together with Proposition 9 and (57)-(60), it follows in turn that

In the sequel, we study the property of the asymptotic expansion of a function in the class at a real point .

Proposition 12. Let and . If, for some , has an asymptotic expansion at of the form: where , then has the angular derivatives of order from 1 to at the real point ; moreover,

Proof. From the asymptotic expansion (66) of at , we deduce Observe that , , and thus (67) holds.

Theorem 11 and Proposition 12 say actually that if, for some , has an asymptotic expansion of the form (66) at , then

Theorem 13. Let and . If, for some , has an asymptotic expansion of the form (66) at , then

Proof. Let and . Then we have In view of the relation , we obtain, for real , Since is a polynomial, we have Therefore, in order to prove (70), we need only to prove By Proposition 3, we have A direct computation leads to By L’Hospital’s rule, together with (69), we have This means that In a similar way, we can verify that Then (74) holds, as needed.

From Theorems 7 and 13, we obtain an interesting result about the number of negative eigenvalues of the Hankel matrix constructed with the partial coefficients of the asymptotic expansion of at .

Theorem 14. Let and . If, for some , has an asymptotic expansion of the form (66) at real , then , in which .

Proof. By Theorem 13, we have and therefore is a real Hankel matrix of order . On the other hand, it follows from Theorem 7 that . Since the eigenvalues of a matrix are continuous functions of its entries, we obtain .

Finally, we consider two properties of the asymptotic expansion of at infinity.

Theorem 15. If, for some , has an asymptotic expansion at infinity: where , then , where .

Proof. Let . Then . It follows from (81) that has an asymptotic expansion at the point 0 of the form: where By Theorem 14, we obtain further that , in which . Observe that , where . Therefore, , so that , as desired.

Theorem 16. Let have an asymptotic expansion at infinity of the form: where , , and let . Then, for an arbitrary and mutually different points , where

Proof. Let . Then . For an arbitrary and mutually different points , are also mutually different points. Then, for each , Observe that . Thus we infer
On the other hand, it follows from (84) that Then we have where is a real Hankel matrix of order . Choose which is a nonsingular matrix of order . A direct computation leads to Therefore, .

4. An Explicit Formula of the Negative Index of Real Rational Functions

In this section, we give an explicit formula of the negative index of real rational functions by decomposing each real rational function into the sum of a real polynomial and a strictly proper real rational function.

From definition of a generalized Nevanlinna function, we can prove that each real polynomial is a generalized Nevanlinna function in the class with some negative index determined by the following lemma (see, e.g., [1], [4, Section 3]).

Lemma 17. Let be a real polynomial. Then , where

Proof. Obviously, the real polynomial is holomorphic in such that . Then has no poles in and no generalized poles of nonpositive type in . In what follows we will determine the multiplicity of the point as a generalized pole of nonpositive type of .(1)If and , then so that, by definition, the point is a generalized pole of nonpositive type of with multiplicity .(2)If and , then so that, by definition, the point is a generalized pole of nonpositive type of with multiplicity .(3)If , then so that, by definition, the point is a generalized pole of nonpositive type of with multiplicity .
Since the total multiplicity of the poles of in and the generalized poles of nonpositive type of in is equal to , the multiplicity of the point as a generalized pole of nonpositive type of is equal to . Hence the equality (92) holds.

Let and . Daho and Langer [9, Proposition ] proposed a sufficient condition for mentioned in introduction part. In view of this result, we can obtain a modified version of that sufficient condition (see also [1, Satz 1.13]).

Lemma 18. Let and have no common poles in and no common generalized poles of nonpositive type in and satisfy the following two conditions:(1)if is a generalized pole of nonpositive type of , then ,    ;(2)if the point is a generalized pole of nonpositive type of , then ,    . Then . Moreover, the poles of in and the generalized poles of nonpositive type of in are also the poles of in and the generalized poles of nonpositive type of in with the same multiplicities.

Proof. Without loss of generality, we need only to verify that Lemma 18 is valid in the case of and . If is a pole of of multiplicity , then is holomorphic at , and hence is a pole of of multiplicity .
If is a generalized pole of nonpositive type of with multiplicity , then or is equal to and . It follows from condition (1) that Thus, by definition, is a generalized pole of nonpositive type of with multiplicity .
If the point is a generalized pole of nonpositive type of with multiplicity , then or is equal to and . It follows from condition (2) that Thus the point is a generalized pole of nonpositive type of with multiplicity by definition.
The above analysis shows that the poles of in and the generalized poles of nonpositive type of in are also the poles of in and the generalized poles of nonpositive type of in with the same multiplicities. However, since and have no common poles in and no common generalized poles in , the total multiplicity of poles of in and the generalized poles of nonpositive type of in is greater than or equal to , and hence with . On the other hand, as stated in the introduction part, we have . Thus .

To describe the negative index of a strictly proper real rational function, we have to consider the Bézout matrix of a pair of polynomials (see, e.g., [17–19]) and the generating functions of a Hankel matrix.

Let and be real polynomials with . We define the Bézout matrix of polynomials and as the by matrix generated by Clearly, we have that and .

Let be given. Let be a real polynomial of degree , and let be a real polynomial of degree not more than such that gcd. If the strictly proper rational function has an asymptotic expansion at the point as follows: then is called a generating function, of McMillan degree , of the Hankel matrix , and in this case that Hankel matrix is also represented as .

It is well-known that the Bézout matrix and the Hankel matrix have the following relation; see, for example, [17, 20].

Lemma 19. Let and be two real polynomials such that the rational function has an asymptotic expansion at the point as Then where

In the following, we give the result on the negative index of a strictly proper real rational function and its detailed proof.

Theorem 20. Let be a real polynomial of degree , and let be a real polynomial of degree not more than . If has an asymptotic expansion at the point as follows: then , where and .

Proof. Observe that and, for an arbitrary and distinct points , where Thus we obtain that , and hence and .
On the other hand, since has an asymptotic expansion of the form (103), by Theorem 15, we have , with , where . Moreover, Lemma 19 provides that , where is invertible, so that , and hence . Therefore, we have with .

For an arbitrary real rational function , where and are real polynomials such that gcd and . By the Euclidean division of polynomials, has an expression of the form , in which or ; hence, we have , where or is the form , with .

If , then, by Theorem 20, the negative index of is equal to the negative inertia index of the by Hankel matrix generated by the function .

If , we define as the polynomial part of and as the strictly proper rational function part of . In this case, the negative index of is given by use of Lemma 17 and the negative index of is derived by use of Theorem 20. Therefore, by Lemma 18, the negative index of a real rational function is the sum of the negative indexes of and . Thereby, we obtain an explicit formula of the negative index of real rational functions.

Theorem 21. Let be a real rational function, where and are real polynomials with gcd and . Then(1)If , then , where .(2)If , then , where where is the strictly proper rational function part of .

5. Conclusion

In this paper, we consider some propositions on generalized Nevanlinna functions with detailed proofs. We prove that the Hermitian matrix is congruent to the Hermitian matrix , given in (29). This result is quite important in the proof of an equivalent definition of a generalized Nevanlinna function (see Theorem 7), because it can be used to verify the fact that the negative inertia index of every Hermitian generalized Loewner matrix generated by a generalized Nevanlinna function of the class does not exceed . We also obtain that the negative inertia index of the Hankel matrix constructed with the partial coefficients of the asymptotic expansions of a generalized Nevanlinna function of the class at a real point or at infinity is less than or equal to , and the same result holds also for the Loewner matrix defined by (86). Finally, based on the relations among Loewner, Bézout, and Hankel matrices, we give an explicit formula of the negative index of real rational functions. These results stated in this paper will be applied to solve the indefinite Hamburger moment problems and the multiple Nevanlinna-Pick interpolation problems for the class in subsequent work.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (nos. 11071017 and 11271045) and the Program for New Century Excellent Talents in University.