Neumann gradient estimate for nonlinear heat equation under integral Ricci curvature bounds

1.   Introduction

Throughout this paper, we write $\mathbb C^{p\times q}$ ($\mathbb R^{p\times q}$, resp.) for the set of all $p\times q$ complex (real, resp.) matrices. We use the symbol $\mathbb C^+$ to stand for the open upper half complex plane. For a Hermitian matrix $A = A^*\in\mathbb C^{p\times p}$, the number of negative eigenvalues (counting algebraic multiplicities) of $A$ is denoted by $\nu(A)$. For convenience, the symbol $\mathbb R_n[z]$ ($\mathbb R_n^0[z]$, resp.) represents the set of all real coefficient polynomials of degree at most $n$ (real coefficient polynomials of degree $n$, resp.) of the variable $z$. For a rational function $r(z)$ of the form $r(z) = p(z)/q(z)$, in which $p(z), q(z)$ are nonzero complex polynomials such that $\gcd(p(z), q(z)) = 1$, the McMillan degree of $r(z)$ is defined by $\deg r(z) = \max\{\deg p(z), \deg q(z)\}$.

Let $f(z)$ be a function meromorphic on $\mathbb C\setminus \mathbb R$. The domain of $f(z)$ is denoted by $D(f)$. For a nonnegative integer $\kappa$, $f(z)$ is called a generalized Nevanlinna function with negative index $\kappa$ if the following statements hold: (i) $f(z)$ satisfies the symmetry condition $f(\overline z) = \overline{f(z)}$; (ii) for each choice of a positive integer $m$ and $m$ distinct points $z_{1}, \ldots, z_{m}\in \mathbb{C}^{+}\cap D(f)$, we have

and, for some particular choice, the equality in (1.1) holds, where

is a Hermitian Loewner matrix. We denote by $\mathcal{N}_{\kappa}$ the class of all generalized Nevanlinna functions with negative index $\kappa$.

The indefinite Hamburger moment problem in the class $\mathcal{N}_{\kappa}$ (short for the HM($\mathcal{N}_{\kappa}$) problem) can be formulated in the following manner: Given a sequence of real numbers $s_0, \cdots, s_{2n-2}$, it is required to find all functions $f(z)\in \mathcal{N}_{\kappa}$ such that the following asymptotic expansion at infinity

holds when $z$ tends to $\infty$ in the sector $\pi_\epsilon(0) = \{z\in\mathbb{C}\mid \epsilon{\leq} \arg z{\leq} \pi-\epsilon\}\ (0 < \epsilon < \pi/2)$.

The classical Hamburger moment problem (e.g., ^[1,15]) is identical to the HM($\mathcal{N}_{\kappa}$) problem with $\kappa = 0$. In comparison with the classical case, the HM($\mathcal{N}_{\kappa}$) problem with $\kappa > 0$ seems to be much more complicated. In 2003, Derevyagin ^[7] applied the step-by-step Schur algorithm to give a description of the solutions of the HM($\mathcal{N}_{\kappa}$) problem when the Hankel matrix

determined by the asymptotic expansion (1.2) is nonsingular. In 2012, Derkach et al. ^[8] gave the solvability criterion of the HM($\mathcal{N}_{\kappa}$) problem and a complete parametrization description of the solutions by using the same algorithm. In this paper, we derive the solvability criterion for the HM($\mathcal{N}_{\kappa}$) problem by a new approach, which is more algebraic and different from the existing methods. As a by-product of this approach, we obtain a concrete rational solution of the HM($\mathcal{N}_{\kappa}$) problem with the least McMillan degree when the solvability conditions are met.

We remark that, starting from the Hankel matrix ${\bf H}$ given by (1.3), we can derive the solvability criterion of the HM($\mathcal{N}_{\kappa}$) problem and the concrete formula of the solutions when the solvability conditions are met. For this reason, we say ${\bf H}$ is the Hankel matrix of the HM($\mathcal{N}_{\kappa}$) problem. Moreover, the HM($\mathcal{N}_{\kappa}$) problem is said to be non-degenerate (degenerate, resp.) if its Hankel matrix is nonsingular (singular, resp.). In this paper, we divide the HM($\mathcal{N}_{\kappa}$) problem into the non-degenerate case and the degenerate case to derive the solvability criterion by using the structural characteristics of the Hankel matrix ${\bf H}$, such as the characteristic degrees and characteristic polynomial quadruple (see, e.g., ^[4,5]), the quasidirect decomposition (see, e.g., ^[9]), and the relation among the Hankel, Loewner, Bezout and some other structured matrices (see, e.g., ^[4,6,11]).

A brief synopsis of this paper is as follows. In Section 2, we introduce the characteristic degrees, characteristic polynomial quadruple and quasidirect decomposition of the Hankel matrix of the HM(${\cal N}_\kappa$) problem, and we list some basic results about these structural characteristics without proofs. In Section 3, we first recall some known properties of the generalized Nevanlinna functions in ^[17] given by the first two authors of this paper and their collaborators, and then, we prove several new properties by using the structural characteristics of the Hankel matrix and the relation among the Hankel, Loewner, Bezout and some other structured matrices. The last section is devoted to the solvability criterion of the HM($\mathcal{N}_{\kappa}$) problem and a concrete rational solution with the least McMillan degree when the solvability conditions are met.

2.   Structural characteristics of the Hankel matrix of the HM(${\cal N}_\kappa$) problem

Let ${\bf H} = (s_{i+j})_{i, j = 0}^{n-1}\in \mathbb{R}^{n\times n}$ be the Hankel matrix of the HM(${\cal N}_\kappa$) problem. The first and the second characteristic degrees of ${\bf H}$ are defined by $n_1 = {\rm rank}\, {\bf H}$, $n_2 = 2n-n_1$, respectively. Clearly, $n_1{\leq} n{\leq} n_2$. For a pair of positive integers $k, l$ such that $k+l = 2n$, we write $\mathbf{H}_{kl} = (s_{i+j})_{i, j = 0}^{k-1, l-1}$. In particular, ${\bf H} = {\bf H}_{nn}$. It follows from ^[13] that $\mbox{rank}\ {\bf H}_{kl} = \min\{k, l, n_1\}$. We use the symbol $\mathcal A_l$ to stand for the subspace of ${\mathbb{R}}_{l-1}[z]$:

In the case $n_1 = n_2 = n$, ${\bf H}$ is nonsingular, and there exists a unique monic polynomial $p(z)\in {\mathbb{R}}_{n-1}[z]$ and a polynomial $q(z)\in {\mathbb{R}}^0_{n}[z]$, forming a basis of the space $\mathcal A_{n+1}$. In the case $n_1 < n < n_2$, ${\bf H}$ is singular and there exists uniquely a monic polynomial $p(z)\in {\mathbb{R}}_{n_1}[z]$, forming a basis of $\mathcal A_{n_1+1}$ and, moreover, a polynomial $q(z)\in {\mathbb{R}}_{n_2}[z]$ such that $p(z), zp(z), \ldots, z^{n_2-n_1} p(z), q(z)$ forms a basis of the space $\mathcal A_{n_2+1}$. For convenience, in this paper we always assume that either $\deg p(z) = n_1$ and $\deg q(z) < n_2$ or $\deg p(z) < n_1$ and $\deg q(z) = n_2$. Such a pair of polynomials $p(z)$ and $q(z)$ are referred to as the first and the second characteristic polynomials of ${\bf H}$, respectively.

Let $p(z) = p_{n_1}z^{n_1}+\cdots+p_0$ and $q(z) = q_{n_2}z^{n_2}+\cdots+q_0$. We define two real coefficient polynomials $\gamma(z)$ and $\delta(z)$ by

Hereafter, $[p(z), q(z), \gamma(z), \delta(z)]$ is called the characteristic polynomial quadruple of ${\bf H}$. Such a quadruple, together with the first and second characteristic degrees, plays an important role in our discussion. By using the definitions, we check easily that the following asymptotic expansions at infinity hold:

The following lemma comes as a direct consequence of the definitions and the last two equations (e.g., ^[4]).

Lemma 2.1. Let $n_1$ and $[p(z), q(z), \gamma(z), \delta(z)]$ be the first characteristic degree and the characteristic polynomial quadruple of the Hankel matrix ${\bf H} = (s_{i+j})_{i, j = 0}^{n-1}$ given by (1.3), respectively. Then,

where $\sigma$ is a nonzero constant. Moreover, if $\deg p(z) = n_1$, then $f_{\gamma, p}(z) = -\gamma(z)/p(z)$ admits the following asymptotic expansion at infinity:

if $\deg p(z) < n_1$ then $f_{\delta, q}(z) = -\delta(z)/q(z)$ admits the following asymptotic expansion at infinity:

Remark 2.2. By (2.1) and the fact that $\deg p(z)+\deg q(z){\leq} 2n-1$, we can show that the asymptotic expansions (2.2) and (2.3) cannot hold simultaneously. This means that if the asymptotic expansion (2.2) ((2.3), resp.) holds, then $\deg p(z) = n_1$ ($\deg p(z) < n_1$, resp.).

Now, we introduce the quasidirect decomposition of a singular Hankel matrix. Let ${\bf H} = (s_{i+j})_{i, j = 0}^{n-1}$ given by (1.3) be singular. We say ${\bf H}$ is a proper Hankel matrix if $\Delta_{n_1}\neq 0$ and $\Delta_{k} = 0\ (k = n_1+1, \cdots, n)$, in which $\Delta_i$ stands for the $i$th leading principle minor of ${\bf H}$. Moreover, we say that ${\bf H}$ is a degenerate Hankel matrix if $s_0 = s_1 = \cdots = s_{n-1} = 0$. By definitions, a $n\times n$ zero matrix is both a proper Hankel matrix and a degenerate Hankel matrix. In ^[9], Fielder showed that under certain conditions, each singular Hankel matrix can be uniquely decomposed into the sum of a proper Hankel matrix and a degenerate Hankel matrix.

Lemma 2.3. Let ${\bf H} = (s_{i+j})_{i, j = 0}^{n-1}$ given by (1.3) be singular. Then, ${\bf H}$ has a unique decomposition of the form:

in which ${\bf H}^{\rm p}$ is a proper Hankel matrix and ${\bf H}^{\rm d}$ is a degenerate Hankel matrix.

The formula (2.4) is called the quasidirect decomposition of the Hankel matrix ${\bf H}$. By definition, together with Lemmas 2.1, 2.3 and Remark 2.2, we can give a characterization of the proper Hankel matrix by using the structural characteristics.

Lemma 2.4. Let ${\bf H} = (s_{i+j})_{i, j = 0}^{n-1}$ given by (1.3) be singular, and let $n_1$ and $[p(z), q(z), \gamma(z), \delta(z)]$ be the first characteristic degree and the characteristic polynomial quadruple of ${\bf H}$, respectively. Then, ${\bf H}$ is a proper Hankel matrix if, and only if, one of the following statements holds:

(1) $\deg p(z) = n_1$;

(2) $f_{\gamma, p}(z) = -\gamma(z)/p(z)$ admits the asymptotic expansion (2.2) at infinity.

The following lemma shows that if the Hankel matrix of the HM(${\cal N}_\kappa$) problem is singular, then it is equivalent and congruent to a block diagonal matrix, where the upper left corner and the lower right corner blocks are Hankel matrices whose orders coincide with the ranks of its proper part and degenerate part, respectively, and the other blocks are zero matrices. Such a structural characteristic of the singular Hankel matrix plays an important role in deducing the solvability criterion of the HM(${\cal N}_\kappa$) problem for the degenerate case.

Lemma 2.5. Let ${\bf H} = (s_{i+j})_{i, j = 0}^{n-1}$ be a singular Hankel matrix with the quasidirect decomposition (2.4), let $n_1$ and $p(z) = z^r+p_{r-1}z^{r-1}+\cdots+p_1z+p_0$ be the first characteristic degree and the first characteristic polynomial of ${\bf H}$, respectively and let

Then, $Q$ is a nonsingular matrix satisfying

in which

are nonsingular Hankel matrices of small sizes.

Hereafter, we always assume that the Hankel matrix $\widehat {\bf H}^{\rm p}$ ($\widehat {\bf H}^{\rm d}$, resp.) does not appear in the case $r = 0$ ($r = n_1$, resp.).

3.   Some properties of the generalized Nevanlinna functions

In ^[17], Song et al. presented an equivalent definition of the generalized Nevanlinna function with negative index $\kappa$ in terms of the generalized Loewner matrix. On the basis of this definition, they have shown several properties of such a kind of function. Here, we list two interesting properties of them. The first property is stated as follows, which gives a necessary condition for the HM(${\cal N}_\kappa$) problem to have a solution.

Lemma 3.1. ^[17] Let ${\bf H} = (s_{i+j})_{i, j = 0}^{n-1}$ be the Hankel matrix of the HM(${\cal N}_\kappa$) problem. If $f(z)\in \mathcal N_{\kappa}$ admits the asymptotic expansion (1.2) at infinity, then $\kappa{\geq} \nu({\bf H})$.

The second property is actually a generalization of the first one. Starting from this property, we can derive the solvability criterion of the HM(${\cal N}_\kappa$) problem in the degenerate case.

Lemma 3.2. ^[17] If $f(z)\in \mathcal N_{\kappa}$ admits the asymptotic expansion (1.2) at infinity, then for each positive integer $m$ and $m$ distinct points $z_1, \ldots, z_m\in \mathbb C^+\cap D(f)$,

in which

${\bf H} = (s_{i+j})_{i, j = 0}^{n-1}$ is the Hankel matrix of the HM(${\cal N}_\kappa$) problem and

for arbitrary $z\in \mathbb{C}^+\cap D(f)$.

We observe that ${\bf H}$ is a principle submatrix of $L_f(z_1, \cdots, z_m)$. By the interlacing relation between the eigenvalues of a Hermitian matrix and its principle submatrices (e.g., [14, Theorem 4.3.28]), we have $\nu({\bf H}){\leq} \nu(L_f(z_1, \cdots, z_m))$, then Lemma 3.1 is a direct consequence of Lemma 3.2. If the Hankel matrix ${\bf H}$ of the HM(${\cal N}_\kappa$) problem is singular, by Lemma 2.3 it has a unique quasidirect decomposition. In that case, applying Lemmas 2.5 and 3.2, we can prove the following property of the functions in the class ${\cal N}_\kappa$.

Theorem 3.3. Let $\mathbf{H} = (s_{i+j})_{i, j = 0}^{n-1}$ given by (1.3) be singular, $n_1$ and $[p(z), q(z), \gamma(z), \delta(z)]$ be the first characteristic degree and the characteristic polynomial quadruple of ${\bf H}$, respectively and let $f(z)\in\mathcal N_{\kappa}$ admit the asymptotic expansion (1.2) at infinity. Then, for each positive integer $m$ and $m$ distinct points $z_1, \ldots, z_m\in \mathbb C^+\cap D(f)$, the structured matrix ${L}_{f}(z_{1}, \cdots, z_{m})$ defined by (3.1) and (3.2) is equivalent and congruent to

where $\widehat {\mathbf{H}}^{\rm p}$, $\widehat {\mathbf{H}}^{\rm d}$ are the same as in (2.7), and

Proof. Here, we give a proof only for the case $m = 1$. The proof of the case $m > 1$ is completely analogous and, thus, omitted. Let $p(z) = z^r+p_{r-1}z^{r-1}+\cdots+p_1z+p_0$ and $\widetilde {Q} = {{{\rm{diag}}}}(Q, 1)$, in which $Q$ is given by (2.5). Clearly, $\widetilde {Q}$ is also a nonsingular matrix. It follows from Lemma 2.5 and (2.6) that $\widetilde {L}_f(z_1, \cdots, z_m) = \widetilde {Q}L_f(z_1, \cdots, z_m) \widetilde {Q}^*$ is of the form (3.3), in which

By a direct calculation, we have

and the $(k+1)$-th element of $F$ is

then (3.4) holds. Therefore, we complete the proof of Theorem 3.3. □

To derive the solvability criterion of the HM(${\cal N}_\kappa$) problem, we need some properties of the rational generalized Nevanlinna function. To introduce these properties, we recall the concept of the Bezout matrix (see, e.g., ^[10,16]). For a pair of complex polynomials $a(z), b(z)$ with the maximal degree $n$, the Bezout matrix ${\bf B}(a, b)$ is defined by the bilinear form

It is well known that the Bezout matrix has many applications in the theory of system and control (e.g., ^[2,3,12]). The following result shows that each real rational function is a generalized Nevanlinna function, whose negative index coincides with the number of negative eigenvalues of the Bezout matrix of its denominator and numerator polynomials.

Theorem 3.4. Let $a(z)\in \mathbb{R}_n^0[z]$, $b(z)\in\mathbb{R}_n[z]$. Then, the real rational function $f_{b, a}(z) = -b(z)/a(z)\in{\cal N}_\kappa$, in which $\kappa = \nu({\bf B}(a, b))$.

Proof. Since $a(z), b(z)$ are real coefficient polynomials, $f_{b, a}(z)$ is meromorphic in $\mathbb{C}\setminus \mathbb{R}$ such that for each $z\in D(f_{b, a})$, $\overline{z}\in D(f_{b, a})$ and $f_{b, a}(\overline{z}) = \overline{f_{b, a}(z)}$. Moreover, for each choice of a positive integer $m$ and $m$ distinct points $z_1, \cdots, z_m\in \mathbb{C}^+ \cap D(f_{b, a})$, we have

where $\Lambda = {\rm diag}(a(z_{1})^{-1}, \cdots, a(z_{m})^{-1})$ is a nonsingular diagonal matrix and $V = (z_i^{j-1})_{i, j = 1}^{m, n}$ is a Vandermonde matrix. This implies that $\nu(P_{f_{b, a}}(z_{1}, \cdots, z_{m}))\leq\nu({\bf B}(a, b))$. Particularly, in the case of $m = n$, $V$ is also a nonsingular matrix and, thus, $\nu(P_{f_{b, a}}(z_{1}, \cdots, z_n)) = \nu({\bf B}(a, b))$. By the definition of generalized Nevanlinna functions in the class ${\cal N}_\kappa$, we have $f_{b, a}(z) = -b(z)/a(z)\in{\cal N}_\kappa$, in which $\kappa = \nu({\bf B}(a, b))$, then the proof of Theorem 3.4 is completed. □

We remark that there are many interesting connections between Bezout and Hankel matrices. The following lemma shows that the Bezout matrix ${\bf B}(a, b)$ in Theorem 3.4 is equivalent and congruent to a real Hankel matrix generated by the rational function $b(z)/a(z)$ (see, e.g., ^[6,11] for the general case).

Lemma 3.5. Let $a(z) = \sum_{i = 0}^na_iz^i\ (a_n\neq 0)$ and $b(z) = \sum_{i = 0}^{n-1}b_iz^i$ be real coefficient polynomials, and let the asymptotic expansion of $b(z)/a(z)$ at infinity be of the form

Then, ${\bf B}(a, b) = S(a){\bf H}(a, b)S(a)$, in which ${\bf H}(a, b) = (h_{i+j})_{i, j = 0}^{n-1}$ and

is a nonsingular real symmetric matrix.

By Lemma 3.5, the negative index $\kappa$ in Theorem 3.4 can be formulated in terms of the number of negative eigenvalues of the Hankel matrix ${\bf H}(a, b) = (h_{i+j})_{i, j = 0}^{n-1}$.

Corollary 3.6. Let $a(z), b(z)$ and ${\bf H}(a, b)$ be the same as in Lemma 3.5. Then, the real rational function $f_{b, a}(z) = -b(z)/a(z)\in {\cal N}_\kappa$, in which $\kappa = \nu({\bf H}(a, b))$.

4.   Solvability criterion for the HM(${\cal N}_\kappa$) problem

In this section, we apply the structural characteristics of the Hankel matrix of the HM(${\cal N}_\kappa$) problem and the properties of the generalized Nevanlinna functions to deduce the solvability criterion of the HM($\mathcal{N}_{\kappa}$) problem and a concrete rational solution with the least McMillan degree for both the non-degenerate and degenerate cases. We first derive the solvability criterion for the non-degenerate HM(${\cal N}_\kappa$) problem.

Theorem 4.1. Let ${\bf H} = (s_{i+j})_{i, j = 0}^{n-1}$ given by (1.3) be nonsingular. Then, the HM($\mathcal{N}_{\kappa}$) problem is solvable if, and only if, $\kappa{\geq} \nu({\bf H})$.

Proof. The "only if" part is a direct consequence of Lemma 3.1. For the proof of the "if" part, we consider two cases.

Case I: $\kappa = \nu({\bf H})$. Let $n_1, n_2$ and $[p(z), q(z), \gamma(z), \delta(z)]$ be the first characteristic degree, the second characteristic degree and the characteristic polynomial quadruple of ${\bf H}$, respectively. Since ${\bf H}$ is nonsingular, we have $n_1 = n_2 = n$, $\deg p(z) < n_1 = n$ and $\deg q(z) = n$. By Lemma 2.1, the rational function $f_{\delta, q}(z) = -\delta(z)/q(z)$ admits the asymptotic expansion (1.2) at infinity, and moreover, ${\bf H} = {\bf H}(q, \delta)$. On the other hand, by Corollary 3.6, we obtain that $f_{\delta, q}(z)\in {\cal N}_{\kappa'}$, in which $\kappa' = \nu({\bf H}(q, \delta)) = \nu({\bf H}) = \kappa$. Then, $f_{\delta, q}(z)$ is a solution of the non-degenerate HM(${\cal N}_\kappa$) problem.

Case II: $\kappa > \nu({\bf H})$. For convenience, we denote by ${\bf H}_n = {\bf H}$ and $m = \kappa-\nu({\bf H}) > 0$. Define a sequence of Hankel matrices recursively by

in which

Then, ${\bf H}_{n+m} = (s_{i+j})_{i, j = 0}^{n+m-1}$ is equivalent and congruent to ${{{\rm{diag}}}}({\bf H}, -I_m)$, which implies that ${\bf H}_{n+m}$ is a nonsingular Hankel matrix and $\nu({\bf H}_{n+m}) = \nu({\bf H})+m = \kappa$. Let $[u(z), v(z), \alpha(z), \beta(z)]$ be the characteristic polynomial quadruple of ${\bf H}_{n+m}$. Then $\deg u(z) < n+m$ and $\deg v(z) = n+m$. According to the analysis in Case I, we have that $f_{\beta, v}(z)\in {\cal N}_\kappa$ and the asymptotic expansion of $f_{\beta, v}(z)$ at infinity is of the form:

then $f_{\beta, v}(z)$ is a solution of the HM(${\cal N}_\kappa$) problem. Summarizing the analysis above, we complete the proof of the "if" part. □

From the proof of Theorem 4.1, we obtain immediately a concrete rational solution of the non-degenerate HM(${\cal N}_\kappa$) problem with the least McMillan degree when the solvability conditions are met.

Theorem 4.2. Let $\mathbf{H} = (s_{i+j})_{i, j = 0}^{n-1}$ given by (1.3) be nonsingular. If $\kappa{\geq} \nu({\bf H})$, then $r(z) = -\beta(z)/v(z)$ is a rational solution with the least McMillan degree among all rational solutions of the HM(${\cal N}_\kappa)$ problem, in which $[u(z), v(z), \alpha(z), \beta(z)]$ is the characteristic quadruple of the Hankel matrix ${\bf H}_{n+\kappa-\nu({\bf H})}$ defined recursively by (4.1)–(4.2).

Proof. Let $m = \kappa-\nu({\bf H}){\geq} 0$. We can easily see from the proof of Theorem 4.1 that $r(z)$ presented in Theorem 4.2 is a rational solution of the HM(${\cal N}_\kappa)$ problem with McMillan degree $n+m$. It remains to prove that the HM(${\cal N}_\kappa)$ problem has not any rational solution whose McMillan degree is less than $n+m$. If there exists a rational solution $s(z)$ of the non-degenerate HM(${\cal N}_\kappa)$ problem such that $\deg s(z) < n+m$, then $s(z) = b(z)/a(z)$, in which $a(z), b(z)$ are two co-prime real polynomials satisfying $\deg b(z) < \deg a(z) = t < n+m$. Let $a(z) = a_0+a_1z+\cdots+a_{t}z^{t}$ and

in which $s_i' = s_i$, $i = 0, 1, \cdots, 2n-2$. If $t < n$, then ${\bf H}(a_0, \cdots, a_t, 0, \cdots, 0)^{\rm T} = 0$. It contradicts to the nonsingularity of ${\bf H}$. If $n{\leq} t < n+m$, we denote by ${\bf H}_t' = (s_{i+j}')_{i, j = 0}^{t-1}$. In this case, ${\bf H}$ is a nonsingular principle submatrix of ${\bf H}_t'$, and then $\nu({\bf H}_t'){\leq} \nu({\bf H})+t-n < \nu({\bf H})+m = \kappa$. On the other hand, by Corollary 3.6, we have $\kappa = \nu({\bf H}_t')$. It is a contradiction, so $r(z)$ given in Theorem 4.2 is a rational solution of the HM(${\cal N}_\kappa)$ problem with the least McMillan degree. □

To derive the solvability criterion for the HM(${\cal N}_\kappa$) problem for the degenerate case, we need the following result, which can be verified by a direct computation.

Lemma 4.3. Let $A\in\mathbb{C}^{m\times m}$ be nonsingular, $B = B^*\in\mathbb{C}^{m\times m}$ and

Then, $\nu(C) = m$.

Now, we apply the results above to deduce the solvability criterion of the HM(${\cal N}_\kappa$) problem for the degenerate case.

Theorem 4.4. Let $\mathbf{H} = (s_{i+j})_{i, j = 0}^{n-1}$ given by (1.3) be singular and $n_1$ be the first characteristic degree of ${\bf H}$. Then, the HM($\mathcal{N}_{\kappa}$) problem is solvable if, and only if, one of the following statements holds:

(1) $\kappa = \nu({\bf H})$ and ${\bf H}$ is a proper Hankel matrix;

(2) $\kappa{\geq} \nu({\bf H})+n-n_1$.

Proof. Let $n_1$, $n_2$ and $[p(z), q(z), \gamma(z), \delta(z)]$ be the first characteristic degree, the second characteristic degree and the characteristic polynomial quadruple of ${\bf H}$, respectively, and let $\deg p(z) = r{\leq} n_1$. We first prove the "only if" part. Suppose that $f(z)\in{\cal N}_\kappa$ is a solution of the degenerate HM(${\cal N}_\kappa$) problem. By Lemma 3.1, we have $\kappa{\geq}\nu({\bf H})$. In the case of $\kappa = \nu({\bf H})$, it follows from Lemma 3.2 that for each $z\in \mathbb{C}^+\cap D(f)$, we have $\nu(L_f(z)){\leq}\kappa$, in which

and $C(z)$ is the same as in (3.2). Since ${\bf H}$ is a principle submatrix of $L_f(z)$, we have $\nu(L_f(z)){\geq}\nu({\bf H}) = \kappa$ and thus $\nu(L_f(z)) = \kappa$ for all $z\in \mathbb{C}^+\cap D(f)$. By Theorem 3.3, $L_f(z)$ is equivalent and congruent to

in which $\widehat {\mathbf{H}}^{\rm p}$, $\widehat {\mathbf{H}}^{\rm d}$ are the same as in (2.7), and

We check easily that $\widetilde {L}_f(z)$ is furtherly equivalent and congruent to the following block diagonal matrix:

in which $d(z) = p(z)f(z)+\gamma(z)$ and $e(z) = \overline{e(z)}$. Note that $\nu(L_f(z)) = \nu(\widetilde {L}_f(z)) = \nu(\widehat {L}_f(z)) = \kappa = \nu({\bf H}) = \nu(\widehat {\mathbf{H}}^{\rm p})+\nu(\widehat {\mathbf{H}}^{\rm d})$, then by Lemma 4.3, we have $d(z) = p(z)f(z)+\gamma(z) = 0$, $z\in\mathbb{C}^+\cap D(f)$. This implies that $f(z) = f_{\gamma, p}(z) = -\gamma(z)/p(z)$ for a sufficiently large $|z|$ and $z\in\mathbb{C}^+\cap D(f)$. Since $f(z)$ admits the asymptotic expansion (1.2) at infinity, we have

By Lemma 2.4, we have $\deg p(z) = n_1$ and, thus, ${\bf H} = {\bf H}^{\rm p}$ is a proper Hankel matrix.

When $\kappa > \nu({\bf H})$, $f(z)\not\equiv f_{\gamma, p}(z) = -\gamma(z)/p(z)$. Otherwise, $f_{\gamma, p}(z)\in\mathcal{N}_{{\kappa}}$ and the asymptotic expansion (4.3) holds. In this case, by Lemma 2.4, ${\bf H} = {\bf H}^p$ is a proper Hankel matrix. Moreover, by Corollary 3.6, we have ${\kappa} = \nu({\bf H}(p, \gamma)) = \nu(\widehat {\bf H}^{\rm p}) = \nu({\bf H}^{\rm p}) = \nu(\mathbf{H})$, which contradicts the assumption ${\kappa} > \nu(\mathbf{H})$. We write $m = n-n_1$ and $g(z) = p(z)f(z)+{\gamma}(z)$ for short, then there exist $m$ distinct points $z_1, \cdots, z_m\in \mathbb{C}^+\cap D(f)$ such that $g(z_k)\neq 0$, $k = 1, \cdots, m$. By Lemma 3.2 and Theorem 3.3, we have $\nu(\widetilde {L}_f(z_1, \cdots, z_m)){\leq}\kappa$, in which

We check easily that $\widetilde {L}_f(z_1, \cdots, z_m)$ is equivalent and congruent to the block diagonal matrix ${{{\rm{diag}}}}(\widehat{\mathbf{H}}^{\rm p}, \widehat{\mathbf{H}}^{\rm d}, D)$, where

and $B = B^*\in \mathbb{C}^{m\times m}$. Since ${\rm det}\, A = g(z_1)\cdots g(z_m)\prod_{1\leq i < j\leq m} (z_j-z_i)\neq 0$, $A$ is a nonsingular $m\times m$ matrix. Applying Lemma 4.3, we obtain that $\nu(L_f(z_1, \cdots, z_m)) = \nu(\widehat{\mathbf{H}}^{\rm p})+\nu(\widehat{\mathbf{H}}^{\rm d})+m = \nu(\mathbf{H})+n-n_1$. Therefore, $\kappa{\geq} \nu({\bf H})+n-n_1$, as needed.

Let us turn to prove the "if" part. First we suppose that $\kappa = \nu({\bf H})$ and ${\bf H}$ is a proper Hankel matrix. Then, $\deg p(z) = n_1 < n$, and by Lemma 2.1, $f_{\gamma, p}(z) = -\gamma(z)/p(z)$ admits the asymptotic expansion (1.2) at infinity. By Corollary 3.6, we have $f_{\gamma, p}(z)\in {\cal N}_{\kappa'}$, in which $\kappa' = \nu({\bf H}(p, \gamma)) = \nu(\widehat {\bf H}^{\rm p}) = \nu({\bf H}^{\rm p}) = \nu({\bf H}) = \kappa$, then $f_{\gamma, p}(z)$ is a solution of the degenerate HM(${\cal N}_\kappa$) problem.

Now, we suppose that $\kappa{\geq}\nu({\bf H})+n-n_1$. The proof is divided into two cases.

Case I: $\deg p(z) = n_1$. In this case, by Lemma 2.4, $\mathbf{H} = {\bf H}^{\rm p}$ is a proper Hankel matrix, and the asymptotic expansion (2.2) holds. Assume that

We define an $n_2\times n_2$ Hankel matrix by

Analogous to the proof of Lemma 2.5, there exists a nonsingular matrix $\widetilde {Q}$ of order $n_2$ such that

in which

Clearly, the Hankel matrix ${\mathbf{H}}_{n_2}$ defined by (4.4) and (4.5) is nonsingular. Moreover, by Lemma 4.3, we have $\nu({\mathbf{H}}_{n_2}) = \nu({\mathbf{H}}_{n_1})+n-n_1 = \nu({\mathbf{H}})+n-n_1\leq {\kappa}$, then by Theorem 4.1, there exists a function $f(z)\in{\cal N}_\kappa$ such that the following asymptotic expansion at infinity

holds when $z$ tends to $\infty$ in the sector $\pi_\epsilon(0)$. This means that $f(z)$ is a solution of the degenerate HM(${\cal N}_\kappa$) problem.

Case II: $\deg p(z) < n_1$. By Lemma 2.1, $f_{{{{\rm{d}}}}, q}(z) = -\delta(z)/q(z)$ admits the asymptotic expansion (2.3) at infinity. We assume that

We define an $n_2\times n_2$ Hankel matrix by

Analogous to the proof of Lemma 2.5, there exists a nonsingular matrix $\widetilde {Q}$ of order $n_2$ such that

in which $\widehat {\mathbf{H}}^{\rm p}$ and $\widehat {\mathbf{H}}^{\rm d}$ are the same as in (2.7), and

are real Hankel matrices. We check easily that the Hankel matrix ${\bf H}_{n_2}$ defined by (4.6) and (4.7) is nonsingular. Moreover, it is furtherly equivalent and congruent to the following block diagonal matrix:

in which $\widetilde {C}$ is a real symmetric matrix of order $n_2-n$. Applying Lemma 4.3, we obtain that $\nu(\mathbf{H}_{n_2}) = \nu(\Lambda) = \nu(\widehat {\mathbf{H}}^p)+\nu(\widehat {\mathbf{H}}^d)+n-n_1 = \nu(\mathbf{H})+n-n_1{\leq} \kappa$, then by Theorem 4.1, there exists a function $f(z)\in{\cal N}_\kappa$ such that the following asymptotic expansion at infinity

holds when $z$ tends to $\infty$ in the sector $\pi_\epsilon(0)$. This implies that $f(z)$ is a solution of the degenerate HM(${\cal N}_\kappa$) problem, and the proof of Theorem 4.4 is completed. □

From the proofs of Theorems 4.1 and 4.4, we can obtain a concrete rational solution with the least McMillan degree among all rational solutions of the degenerate HM($\mathcal{N}_{\kappa}$) problem, when the solvability conditions are met.

Theorem 4.5. Let $\mathbf{H} = (s_{i+j})_{i, j = 0}^{n-1}$ given by (1.3) be singular, and let $n_1, n_2$ and $[p(z), q(z), \gamma(z), \delta(z)]$ be the first characteristic degree, the second characteristic degree and the characteristic quadruple of ${\bf H}$, respectively.

(1) If $\kappa = \nu({\bf H})$ and ${\bf H}$ is a proper Hankel matrix, then $r(z) = -\gamma(z)/p(z)$ is a rational solution with the least McMillan degree among all rational solutions of the HM($\mathcal{N}_{\kappa}$) problem;

(2) If $\kappa{\geq} \nu({\bf H})+n-n_1$, then $r(z) = -\beta(z)/v(z)$ is a rational solution with the least McMillan degree among all rational solutions of the HM($\mathcal{N}_{\kappa}$) problem, where $[u(z), v(z), \alpha(z), \beta(z)]$ is the characteristic quadruple of the Hankel matrix ${\bf H}_{\kappa+n-\nu({\bf H})} = (s_{i+j})_{i, j = 0}^{\kappa+n-\nu({\bf H})-1}$ defined by (4.4) if $\kappa+n-\nu({\bf H}) = n_2$ and $\deg p(z) = n_1$, by (4.6) if $\kappa+n-\nu({\bf H}) = n_2$ and $\deg p(z) < n_1$, and defined recursively by

if $\kappa+n-\nu({\bf H}) > n_2$.

Proof. We see from the proofs of Theorems 4.1 and 4.4 that the rational function $r(z)$ presented in Theorem 4.5 for each case is a solution of the HM($\mathcal{N}_{\kappa}$) problem. Now it remains to prove that such a rational solution has the least McMillan degree among all rational solutions of the HM($\mathcal{N}_{\kappa}$) problem, when the solvability conditions are met. We divide the proof into two cases.

Case I: $\kappa = \nu({\bf H})$ and ${\bf H}$ is a proper Hankel matrix. In this case, $\deg r(z) = \deg p(z) = n_1$ and $\widehat{\bf H}^{\rm p} = (s_{i+j})_{i, j = 0}^{n_1-1}$ is nonsingular. Assume that the HM($\mathcal{N}_{\kappa}$) problem has a rational solution $f(z)$ such that $\deg f(z) < n_1$, then there exist two co-prime real polynomials $a(z), b(z)$ such that $f(z) = b(z)/a(z)$, $\deg a(z){\leq} n_1-1$. Let $a(z) = a_0+a_1z+\cdots+a_{n_1-1}z^{n_1-1}$. In view of the fact that

we have $\widehat{\bf H}^{\rm p}(a_0, a_1, \cdots, a_{n_1-1})^{\rm T} = 0$. It contradicts to the nonsingularity of $\widehat{\bf H}^{\rm p}$, then $r(z) = -\gamma(z)/p(z)$ is a rational solution of the HM($\mathcal{N}_{\kappa}$) problem with the least McMillan degree $n_1$.

Case II: $\kappa{\geq} \nu({\bf H})+n-n_1$. In this case, $\deg r(z) = \deg v(z) = \kappa+n-\nu({\bf H})$. If the HM($\mathcal{N}_{\kappa}$) problem has a rational solution $f(z)$ such that $\deg f(z) = t < \kappa+n-\nu({\bf H})$, then there exist two coprime real polynomials $a(z), b(z)$ such that $f(z) = b(z)/a(z)$ and $\deg a(z) = t$. Let

in which $s_i' = s_i$, $i = 1, \cdots, 2n-2$, and let ${\bf H}_k' = (s_{i+j}')_{i, j = 0}^{k-1}$, $k = 1, 2, \cdots$. By Corollary 3.6, we have $\nu({\bf H}_t') = \kappa{\geq} \nu({\bf H})+n-n_1$. If $t < n_2$, then ${\bf H}_{n_2}'$ is singular and, thus, $\nu({\bf H}_t'){\leq} \nu({\bf H}_{n_2}') < \nu({\bf H})+(n_2-n_1)/2 = \nu({\bf H})+n-n_1$. It is a contradiction. Therefore, $t{\geq} n_2$ and ${\bf H}_{n_2}'$ is a principle submatrix of ${\bf H}_t'$. By the interlacing relation between the eigenvalues of a Hermitian matrix and its principle submatrices (see, e.g., [14, Theorem 4.3.28]),

It is also a contradiction, and $r(z) = -\beta(z)/v(z)$ is a rational solution of the HM($\mathcal{N}_{\kappa}$) problem with the least McMillan degree $\kappa+n-\nu({\bf H})$. □

5.   Conclusions

In this paper, we introduced some basic structural characteristics of the Hankel matrix, such as the first and second characteristic degrees, the characteristic polynomial quadruple and the quasidirect decomposition for the singular case, and then, we applied these structural characteristics and the relation among the Hankel, Loewner, Bezout and some other structured matrices to deduce several new properties of the functions in the class ${\cal N}_\kappa$ and the solvability criterion of the HM(${\cal N}_\kappa$) problem for both the non-degenerate and degenerate cases. As a by-product, we simultaneously obtained a rational solution of the HM(${\cal N}_\kappa$) problem with the least McMillan degree when the solvability conditions were met.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The corresponding author is supported by the Scientific Research Fund from Beijing Normal University at Zhuhai (Grant No. 111032119).

Conflict of interest

Yongjian Hu is the Guest Editor of special issue "Matrix theory and its applications" for AIMS Mathematics. Yongjian Hu was not involved in the editorial review and the decision to publish this article. All authors declare no conflicts of interest in this paper.