On the Roots of a Hyperbolic Polynomial Pencil
victorkatsnelson@gmail.com
Abstract
Let $ \nu_0(t),\nu_1(t),\ldots,\nu_n(t)$ be the roots of the equation $ R(z)=t$, where $ R(z)$ is a rational function of the form
| $\displaystyle \begin{eqnarray*} R(z)=z-\sum\limits_{k=1}^n\frac{\alpha_k}{z-\mu_k}, \end{eqnarray*}$ |
$ \mu_k$ are pairwise distinct real numbers, $ \alpha_k> 0,\,1\leq{}k\leq{}n$. Then for each real $ \xi$, the function $ e^{\xi\nu_0(t)}+e^{\xi\nu_1(t)}+\,\cdots\,+e^{\xi\nu_n(t)}$ is exponentially convex on the interval $ -\infty< t< \infty$.
Keywords
Exponentially convex functions
Mathematics Subject Classification
11C99, 26C10, 26C15, 15A22, 42A821 Roots of the Equation $ {R(z)=t}$ as Functions of $ {t}$
In the present paper we discuss questions related to properties of roots of the equation
| $ \begin{equation*} \label{Eqt} R(z)=t \end{equation*}$ | (1.1) |
as functions of the parameter $ t\in\mathbb{C}$, where $ R$ is a rational function of the form
| $ \begin{equation*} \label{RaF} R(z)=z-\sum\limits_{1\leq k \leq n}\frac{\alpha_{k}}{z-\mu_{k}}, \end{equation*}$ | (1.2) |
$ \mu_k$ are pairwise distinct real numbers, $ \alpha_k> 0$, $ 1\leq k\leq n$. We adhere to the enumeration agreement11We assume that $ n \geq 1$.
| $ \begin{equation*} \label{EnAg} \mu_1> \mu_2> \cdots> \mu_n. \end{equation*}$ | (1.3) |
The function $ R$ is representable in the form
| $ \begin{equation*} \label{RR} R(z)=\frac{P(z)}{Q(z)}, \end{equation*}$ | (1.4) |
where
| $\displaystyle \begin{eqnarray*}\label{QDe} &Q(z)=(z-\mu_1)\cdot(z-\mu_2)\cdot\,\cdots\,\cdot(z-\mu_n), \end{eqnarray*}$ | (1.5) |
| $\displaystyle \begin{eqnarray*} \label{PDe} &P(z)\stackrel{\text{def}}{=}R(z)\cdot{}Q(z) \end{eqnarray*}$ | (1.6) |
are monic polynomials of degrees
| $ \begin{equation*}\label{Deg} \text{deg}\,P=n+1,\quad \text{deg}\,Q=n. \end{equation*}$ | (1.7) |
Since $ P(\mu_k)=-\alpha_k Q^{\prime}(\mu_k)\not=0$, the polynomials $ P$ and $ Q$ have no common roots. Thus the ratio in the right hand side of (1.4) is irreducible. The Eq. (1.1) is equivalent to the equation
| $ \begin{equation*} \label{EqEq} P(z)-tQ(z)=0. \end{equation*}$ | (1.8) |
Since the polynomial $ P(z)-tQ(z)$ is of degree $ n+1$, the latter equation has $ n+1$ roots for each $ t\in\mathbb{C}$.
The function $ R$ possess the property
| $ \begin{equation*} \label{NePr} \text{Im}\,R(z)\big/\text{Im}\,z> 0 \quad \ \text{if}\quad \text{Im}\,z\not= 0. \end{equation*}$ | (1.9) |
Therefore if $ \text{Im}\,t> 0$, all roots of the equation (1.1), which is equivalent to the Eq. (1.8), are located in the half-plane $ \text{Im}\,z> 0$. Some of these roots may be multiple.
However if $ t$ is real, all roots of the Eq. (1.1) are real and simple, i.e. of multiplicity one. Thus for real $ t$, the Eq. (1.1) has $ n+1$ pairwise distinct real roots $ \nu_k(t)$: $ \nu_0(t)> \nu_1(t)> \cdots> \nu_{n-1}(t)> \nu_n(t)$. Moreover for each real $ t$, the poles $ \mu_k$ of the function $ R$ and the roots $ \nu_k(t)$ of the Eq. (1.1) are interlacing:
| $\displaystyle \begin{eqnarray*}\label{InSp} \nu_0(t)> \mu_1> \nu_1(t)> \mu_2> \nu_2(t)> \cdots > \nu_{n-1}(t)> \mu_n> \nu_{n}(t), \quad \forall\,t\in\mathbb{R}.\nonumber\\ \end{eqnarray*}$ | (1.10) |
In particular for $ t=0$, the roots $ \nu_k(0)=\lambda_k$ of the Eq. (1.1) are the roots of the polynomial $ P$:
| $\displaystyle \begin{align} \label{PRo} & P(z)=(z-\lambda_0)\cdot(z-\lambda_1)\cdot\,\,\cdots\,\,\cdot(z-\lambda_n),\\ & \lambda_0> \mu_1> \lambda_1> \mu_2> \lambda_2> \cdots> \lambda_{n-1}> \mu_n> \lambda_{n}. \end{align}$ | (1.11) |
Since $ R^{\prime}(x)> 0$ for $ x\in\mathbb{R},\,x\not=\mu_1,\ldots,\mu_n$, each of the functions $ \nu_k(t),k=0,1,\ldots,n$, can be continued as a single valued holomorphic function to some neighborhood of $ \mathbb{R}$. However the functions $ \nu_k(t)$ can not be continued as single-valued analytic functions to the whole complex $ t$-plane. According to (1.4),
| $ \begin{equation*} \label{DR} R^{\prime}(z)=\frac{P^{\prime}(z)Q(z)-Q^{\prime}(z)P(z)}{Q^2(z)}. \end{equation*}$ | (1.12) |
The polynomial $ P^{\prime}Q-Q^{\prime}P$ is of degree $ 2n$ and is strictly positive on the real axis. Therefore this polynomial has $ n$ roots $ \zeta_1,\ldots,\zeta_n$ in the upper half-plane $ \text{Im}(z)> 0$ and $ n$ roots $ \overline{\zeta_1},\ldots,\overline{\zeta_n}$ in the lower half-plane $ \text{Im}(z)< 0$. (Not all roots $ \zeta_1,\ldots,\zeta_n$ must be distinct.) The points $ \zeta_1,\ldots,\zeta_n$ and $ \overline{\zeta_1},\ldots,\overline{\zeta_n}$ are the critical points of the function $ R$: $ R^{\prime}(\zeta_k)=0,\,R^{\prime}(\overline{\zeta_k})=0,\ 1\leq k\leq n.$ The critical values $ t_k=R(\zeta_k),\,\overline{t_k}=R(\overline{\zeta_k}),\ 1\leq k\leq n,$ of the function $ R$ are the ramification points of the function $ \nu(t)$:
| $ \begin{equation*} \label{RoF} R(\nu(t))=t \end{equation*}$ | (1.13) |
(Even if the critical points $ \zeta^{\prime}$ and $ \zeta^{\prime\prime}$ of $ R$ are distinct, the critical values $ R(\zeta^{\prime})$ and $ R(\zeta^{\prime\prime})$ may coincide.) We denote the set of critical values of the function $ R$ by $ \mathcal{V}$:
| $ \begin{equation*} \label{CrV} \mathcal{V}=\mathcal{V}^{+}\cup\mathcal{V}^{-},\quad \mathcal{V}^{+}=\{t_1,\,\ldots\,,t_n\},\ \mathcal{V}^{-}=\{\overline{t_1},\,\ldots\,,\overline{t_n}\}. \end{equation*}$ | (1.14) |
Not all values $ t_1,\,\ldots\,,t_n$ must be distinct. However $ \mathcal{V}\not=\emptyset$. In view of (1.9), $ \text{Im}\,t_k> 0,\,1\leq k\leq n$. So
| $ \begin{equation*} \label{CrVa} \mathcal{V}^{+}\subset\{t\in\mathbb{C}:\,\text{Im}\,t> 0\},\quad \mathcal{V}^{-}\subset\{t\in\mathbb{C}:\,\text{Im}\,t< 0\}. \end{equation*}$ | (1.15) |
Let $ G$ be an arbitrary simply connected domain in the $ t$-plane which does not intersect the set $ \mathcal{V}$. Then the roots of Eq. (1.1) are pairwise distinct for each $ t\in{}G$. We can enumerate these roots, say $ \nu_0(t),\nu_1(t),\,\ldots\,\nu_n(t)$, such that all functions $ \nu_k(t)$ are holomorphic in $ G$.
The strip $ S_h$,
| $ \begin{equation*} \label{Str} S_h=\{t\in\mathbb{C}:|\text{Im}\,t|< h\},\ \ \text{where} \ \ h=\min\limits_{1\leq k\leq n}\!\text{Im}\,t_k, \end{equation*}$ | (1.16) |
does not intersect the set $ \mathcal{V}$. So $ n+1$ single valued holomorphic branches of the function $ \nu(t)$, (1.13), are defined in the strip $ S_h$. We choose such enumeration of these branches which agrees with the enumeration (1.10) on $ \mathbb{R}$.
From (1.6) and (1.2) it follows that the polynomial $ P$ is representable in the form
| $\displaystyle \begin{eqnarray*} \label{Pa} P(z)=z\,Q(z)-\sum\limits_{k=1}^{n}\alpha_kQ_k(z), \end{eqnarray*}$ | (1.17a) | ||
| where | |||
| $\displaystyle \begin{eqnarray*} \label{Qa} Q_k(z)=Q(z)/(z-\mu_k),\quad k=1,2,\,\ldots\,,n. \end{eqnarray*}$ | (1.17b) | ||
2 Determinant Representation of the Polynomial Pencil $ {P(z)-tQ(z)}$
The polynomial pencil $ P(z)-tQ(z)$ is hyperbolic: for each real $ t$, all roots of the Eq. (1.8) are real.
Using (1.17), we represent the polynomial $ P(z)-tQ(z)$ as the characteristic polynomial $ \det(zI-(A+tB))$ of some matrix pencil, where $ A$ and $ B$ are self-adjoint $ (n+1)\times(n+1)$ matrices, $ \text{rank}\,B=1$. We present these matrices explicitly.
Lemma 2.1.
Let $ A=\|a_{p,q}\|$ and $ B=\|b_{p,q}\|$, $ 0\leq{}p,q\leq{}n$, be $ (n+1)\times(n+1)$ matrices with the entries
| $\displaystyle \begin{eqnarray*} &\displaystyle a_{0,0}=0, \ a_{p,p}=\mu_p \quad \text{for} \ p=1,2,\,\ldots\,,n, \ \ \nonumber\\ &\displaystyle a_{p,q}=0 \quad\ \text{for} \ p=1,2,\,\ldots\,,n,\ q=1,2,\,\ldots\,,n, \ p\not=q, \label{MatA} \nonumber\\ &\displaystyle a_{0,p}=\overline{a_{p,0}} \quad\ \text{for} \ p=1,2,\,\ldots\,,n, \end{eqnarray*}$ | (2.1) |
and
| $\displaystyle \begin{eqnarray*} \label{MatB} b_{0,0}=1, \quad\ \text{all other} \quad\ b_{p,q} \ \quad\text{vanish.} \end{eqnarray*}$ | (2.2) |
Then the equality
| $ \begin{equation*} \label{DeRe} \det(zI-A-tB)=(z-t)\cdot{}Q(z)-\sum\limits_{k=1}^n|a_{0,k}|^2Q_k(z). \end{equation*}$ | (2.3) |
holds.
Proof.
The matrix $ zI-(A+tB)$ is of the form
| $\displaystyle \begin{eqnarray*} zI-(A+tB)= \begin{bmatrix} z-t&\quad -a_{0,1}&\quad -a_{0,2}&\quad \cdots&\quad -a_{0,n-1}&\quad -a_{0,n}\\ -\overline{a_{0,1}}&\quad z-\mu_1&\quad 0&\quad \cdots&\quad 0&\quad 0\\ -\overline{a_{0,2}}&\quad 0&\quad z-\mu_2&\quad \cdots&\quad 0&\quad 0\\ \ldots&\quad \ldots&\quad \ldots&\quad \ldots&\quad \ldots&\quad \ldots\\ -\overline{a_{0,n-1}}&\quad 0&\quad 0&\quad \cdots&\quad z-\mu_{n-1}&\quad 0\\ -\overline{a_{0,n}}&\quad 0&\quad 0&\quad \cdots&\quad 0&\quad z-\mu_n \end{bmatrix} \end{eqnarray*}$ |
We compute the determinant of this matrix using the cofactor formula. $ \square$
Comparing (1.17) and (2.3), we see that if the conditions
| $ \begin{equation*} \label{CruCo} |a_{0,p}|^2=\alpha_p,\quad p=1,2,\,\ldots\,,n \end{equation*}$ | (2.4) |
are satisfied, then the equality
| $ \begin{equation*} \label{CruEq} P(z)-tQ(z)=\det(zI-A-tB) \end{equation*}$ | (2.5) |
holds for every $ z\in\mathbb{C}, t\in\mathbb{C}$.
The following result is an immediate consequence of Lemma 2.1.
Theorem 2.2.
Let $ R$ be a function of the form (1.2), where $ \mu_1,\mu_2,\ldots,\mu_n$ are pairwise distinct real numbers and $ \alpha_1,\alpha_2,\ldots,\alpha_n$ are positive numbers. Let $ Q$ and $ P$ be the polynomials related to the the function $ R$ by the equalities (1.5) and (1.17).
Then the pencil of polynomials $ P(z)-tQ(z)$ is representable as the characteristic polynomial of the matrix pencil $ A+tB$, i.e. the equality (2.5) holds for every $ z\in\mathbb{C}, t\in\mathbb{C}$, where $ B$ is the matrix with the entries (2.2), and the entries of the matrix $ A$ are defined by by (2.1) with
| $ \begin{equation*} \label{UpRo} a_{0,p}=\sqrt{\alpha_p}\,\omega_p, \quad p=1,2,\ldots,n, \end{equation*}$ | (2.6) |
$ \omega_p$ are arbitrary22We will use the freedom in choosing $ \omega_p$ to prescribe signs $ \pm$ to the entries $ a_{0,p}$. complex numbers of absolute value one:
| $ \begin{equation*} \label{Uni} |\omega_p|=1,\quad p=1,2,\ldots,n. \end{equation*}$ | (2.7) |
Corollary 2.3.
Let $ R, A, B$ be the same as in Theorem 2.2. For each $ t\in\mathbb{C}$, the roots $ \nu_0(t),\nu_0(t),\,\ldots\,,\nu_n(t)$ of the Eq. (1.2) are the eigenvalues of the matrix $ A+tB$.
Lemma 2.4.
Let $ R, A, B$ be the same as in Theorem 2.2, $ \nu_0(t),\nu_0(t),\ldots,$ $ \nu_n(t)$ be the roots of the Eq. (1.2) and $ h(z)$ be an entire function. Then the equality
| $ \begin{equation*} \label{TrEqu} \sum\limits_{k=0}^nh(\nu_k(t))={\rm trace}\,\{h(A+tB)\} \end{equation*}$ | (2.8) |
holds for every $ t\in\mathbb{C}$.
Proof.
We refer to Corollary 2.3. If $ \nu$ is an eigenvalue of some square matrix $ M$, then $ h(\nu)$ is an eigenvalue of the matrix $ h(M)$. In (2.8), we interpret the trace of the matrix $ h(A+tB)$ as its spectral trace, that is as the sum of all its eigenvalues. $ \square$
3 Exponentially Convex Functions
Definition 3.1.
A function $ f(t)$ on the interval $ a< t< b$ is said to belong to the class $ W_{a,b}$ if $ f$ is continuous on $ (a,b)$ and if all forms
| $ \begin{equation*} \label{pqf} \sum\limits_{r,s=1}^{N}f(t_r+t_s)\zeta_r\overline{\zeta_s}\quad (N=1,2,3,\ldots\,) \end{equation*}$ | (3.1) |
are non-negative for every choice of complex numbers $ \zeta_1,\zeta_2,\,\ldots\,,\zeta_N$ and for every choice of real numbers $ t_1,t_2,\,\ldots\,,t_N$ assuming that all sums $ t_r+t_s$ are within the interval $ (a,b)$.
The class $ W_{a,b}$ was introduced by [Bernstein1928], see Sect. 15 there. Somewhat later, Widder also introduced the class $ W_{a,b}$ and studied it. Bernstein called functions $ f(x)\in{}W_{a,b}$ exponentially convex.
Properties of the class of exponentially convex functions
- None
-
P 1. If $ f(t)\in{}W_{a,b}$ and $ c\geq0$ is a nonnegative constant, then $ cf(t)\in{}W_{a,b}$.
-
P 2. If $ f_1(t)\in{}W_{a,b}$ and $ f_2(t)\in{}W_{a,b}$, then $ f_1(t)+f_2(t)\in{}W_{a,b}$.
-
P 3. If $ f_1(t)\in{}W_{a,b}$ and $ f_2(t)\in{}W_{a,b}$, then $ f_1(t)\cdot f_2(t)\in{}W_{a,b}$.
-
P 4. Let $ \lbrace f_{n}(t)\rbrace_{1\leq n< \infty}$ be a sequence of functions from the class $ W_{a,b}$. We assume that for each $ t\in(a,b)$ there exists a limit $ f(t)=\lim_{n\to\infty}f_{n}(t)$, and that $ f(t)< \infty\ \forall t\in(a,b)$. Then $ f(t)\in{}W_{a,b}$.
From the functional equation for the exponential function it follows that for each real number $ u$, for every choice of real numbers $ t_1,t_2,\ldots,$ $ t_{N}$ and complex numbers $ \zeta_1$, $ \zeta_2, \ldots, \zeta_{N}$, the equality holds
| $ \begin{equation*} \label{ece} \sum\limits_{r,s=1}^{N}e^{\xi(t_r+t_s)}\zeta_r\overline{\zeta_s}= \bigg|\sum\limits_{p=1}^{N}e^{{\xi}t_p}\zeta_p\,\bigg|^{\,2}\geq 0. \end{equation*}$ | (3.2) |
The inequality (3.2) can be stated as
Lemma 3.2.
For each real number $ \xi$, the function $ e^{\xi t}$ of the variable $ t$ belongs to the class $ W_{-\infty,\infty}$.
The term exponentially convex function is justified by the following integral representation for any function $ f(t)\in{}W_{a,b}$.
Theorem 3.3.
(The representation theorem) For the representation
| $ \begin{equation*} \label{IRep} f(x)=\int\limits_{\xi\in(-\infty,\infty)}e^{\xi{}x}\sigma(d\xi)\quad(a< x< b) \end{equation*}$ | (3.3) |
to be valid, where $ \sigma(d\xi)$ is a non-negative measure, it is necessary and sufficient that $ f(x)\in{}W_{a,b}$.
The proof of the Representation Theorem can be found in [Akhiezer1965] (Theorem 5.5.4), and in [Widder1946] (Chapter 6, Theorem 21).
Corollary 3.4.
The representation (3.3) shows that $ f(x)$ is the value of a function $ f(z)$ holomorphic in the strip $ a< \text{Re}\,z< b$.
4 Herbert Stahl’s Theorem
In the paper [Bessis et al.1975] a conjecture was formulated which is now commonly known as the BMV conjecture:
The BMV conjecture Let $ U$ and $ V$ be Hermitian matrices. Then the function
| $\displaystyle \begin{eqnarray*}\label{TrF} \varphi(t)= \text{trace}\,\{e^{U+tV}\} \end{eqnarray*}$ | (4.1) |
of the variable $ t$ belongs to the class $ W_{-\infty,\infty}$.
If the matrices $ U$ and $ V$ commute, the exponential convexity of the function $ \varphi$, (4.1), is evident. In this case, the sum
| $\displaystyle \begin{eqnarray*} \sum\limits_{r,s=1}^{N}\varphi(t_r+t_s)\zeta_r\overline{\zeta_s}= \text{trace}\,\left\{e^{U/2}\left(\sum\limits_{r=1}^{N}e^{t_rV}\zeta_r\right) \left(\sum\limits_{s=1}^{N}e^{t_sV}\zeta_s\right)^{\ast}(e^{U/2})^{\ast}\right\} \end{eqnarray*}$ |
is non-negative because this sum is the trace of a non-negative matrix. The measure $ \sigma$ in the integral representation (3.3) of the function $ \varphi$, (4.1), is an atomic measure supported on the spectrum of the matrix $ V$.
In the general case, if the matrices $ U$ and $ V$ do not commute, the BMV conjecture remained an open question for longer than 40 years. In 2011, Herbert Stahl proved the BMV conjecture.
Theorem 4.1.
(H. Stahl) Let $ U$ and $ V$ be Hermitian matrices.
Then the function $ \varphi(t)$ defined by (4.1) belongs to the class $ W_{-\infty,\infty}$ of functions exponentially convex on $ (-\infty,\infty)$.
The first arXiv version of Stahl’s Theorem appeared in [Stahl2011], the latest arXiv version—in [Stahl2012], the journal publication—in [Stahl2013].
The proof of Herbert Stahl is based on ingenious considerations related to Riemann surfaces of algebraic functions. In [Eremenko2015], a simplified version of the Herbert Stahl proof is presented.
We present a toy version of Theorem 4.1 which is enough for our goal.
Theorem 4.2.
Let $ U$ and $ V$ be Hermitian matrices. We assume moreover that
-
1.
All off-diagonal entries of the matrix $ U$ are non-negative.
-
2.
The matrix $ V$ is diagonal.
Then the function $ \varphi(t)$ defined by (4.1) belongs to the class $ W_{-\infty,\infty}$.
Proof.
For $ \rho\geq 0$, let $ U_{\rho}=U+\rho{}I$, where $ I$ is the identity matrix. If $ \rho$ is large enough, then all entries of the matrix $ U_{\rho}$ are non-negative. Let us choose and fix such $ \rho$. It is clear that
| $ \begin{equation*} \label{PrF} e^{U+tV}=e^{-\rho}\,e^{U_{\rho}+tV}. \end{equation*}$ | (4.2) |
We use the Lie product formula
| $ \begin{equation*} \label{LPF} e^{U_{\rho}+tV}=\lim_{m\to\infty}(e^{U_{\rho}/m}\,e^{tV/m})^m. \end{equation*}$ | (4.3) |
All entries of the matrix $ e^{U_{\rho}/m}$ are non-negative numbers. Since matrix $ V$ is Hermitian, its diagonal entries are real numbers. Thus
| $\displaystyle \begin{eqnarray*} e^{tV/m}=\text{diag(}e^{tv_1/m},e^{tv_2/m},\ldots,\,e^{tv_m/m}\text{)}, \end{eqnarray*}$ |
where $ v_1,v_2,\ldots,v_m$ are real numbers. The exponentials $ e^{tv_j/m}$ are functions of $ t$ from the class $ W_{-\infty,\infty}$. Each entry of the matrix $ e^{U_{\rho}/m}\,e^{tV/m}$ is a linear combination of these exponentials with non-negative coefficients. According to the properties P1 and P2 of the class $ W_{-\infty,\infty}$, the entries of the matrix $ e^{U_{\rho}/m}\,e^{tV/m}$ are functions of the class $ W_{-\infty,\infty}$. Each entry of the matrix $ (e^{U_{\rho}/m}\,e^{tV/m})^m$ is a sum of products of some entries of the matrix $ e^{U_{\rho}/m}\,e^{tV/m}$. According to the properties P2 and P3 of the class $ W_{-\infty,\infty}$, the entries of the matrix $ (e^{U_{\rho}/m}\,e^{tV/m})^m$ are functions of $ t$ belonging to the class $ W_{-\infty,\infty}$. From the limiting relation (4.3) and from the property P4 of the class $ W_{-\infty,\infty}$ it follows that all entries of the matrix $ e^{U_{\rho}+tV}$ are functions of $ t$ belonging to the class $ W_{-\infty,\infty}$. From (4.2) it follows that all entries of the matrix $ e^{U+tV}$ belong to the class $ W_{-\infty,\infty}$. All the more, the function $ \varphi(t)=\text{trace}\,\{e^{U+tV}\}$, which is the sum of diagonal entries of the matrix $ e^{U+tV}$, belongs to the class $ W_{-\infty,\infty}$. $ \square$
5 Exponential Convexity of the Sum $ e^{\xi\nu_{0}(t)}\,+\cdots+\,e^{\xi\nu_{n}(t)}$
Let $ \xi$ be a real number. Taking $ h(z)=e^{\xi{}z}$ in Lemma 2.4, we obtain
Lemma 5.1.
Let $ R$ be the rational function of the form (1.2), $ \nu_{0}(t),\nu_1(t),\ldots,$ $ \nu_n(t)$ be the roots of the Eq. (1.1). Let $ A$ and $ B$ be the matrices (2.1), (2.6), (2.2) which appear in the determinant representation (2.5) of the matrix pencil $ P(z)-tQ(z)$.
Then the equality
| $ \begin{equation*} \label{trEqu} \sum\limits_{k=0}^{n}e^{\xi\,\nu_k(t)}={\rm trace} \{e^{\xi{}A+t(\xi{}B)}\} \end{equation*}$ | (5.1) |
holds.
Now we choose $ \omega_p$ in (2.6) so that all off-diagonal entries of the matrix $ U=\xi{}A$ are non-negative: if $ \xi> 0$, then $ \omega_p=+1$, if $ \xi< 0$, then $ \omega_p=-1$, $ 1\leq p\leq n$.
Applying Theorem 4.2 to the matrices $ U=\xi{}A, V=\xi{}B$, we obtain the following result
Theorem 5.2.
Let $ R$ be the rational function of the form (1.2), $ \nu_{0}(t),\nu_1(t),$ $ \,\ldots\,, \nu_n(t)$ be the roots of the Eq. (1.1). Then for each $ \xi\in\mathbb{R}$, the function
| $ \begin{equation*} g(t,\xi) \stackrel{\text{def}}{=}\sum\limits_{k=0}^{n}e^{\xi\,\nu_k(t)} \end{equation*}$ | (5.2) |
of the variable $ t$ belongs to the class $ W_{-\infty,\infty}$.
Theorem 5.3.
Let $ f\in{}W_{u,v},\quad \text{where} \ -\infty\leq{}u< v\leq+\infty.$ Let $ R$ be the rational function of the form (1.2), $ \nu_{0}(t),\nu_1(t),$ $ \,\ldots\,, \nu_n(t)$ be the roots of the Eq. (1.1). Assume that for some $ a,b$, $ -\infty\leq{}a< b\leq+\infty$, the inequalities
| $ \begin{equation*} \label{Ine} u< \nu_k(t)< v,\quad a< t< b,\quad\ \ k=0,1,\,\ldots\,,n \end{equation*}$ | (5.3) |
hold.
Then the function
| $ \begin{equation*} \label{SuSu} F(t)\stackrel{\text{def}}{=}\sum\limits_{k=0}^{n}f(\nu_k(t)) \end{equation*}$ | (5.4) |
belongs to the class $ W_{a,b}$.
Proof.
According to Theorem 3.3, the representation
| $ \begin{equation*} f(x)=\int\limits_{\xi\in(-\infty,\infty)}e^{\xi{}x}\sigma(d\xi), \quad \ \forall\,x\in(u,v) \end{equation*}$ |
holds, where $ \sigma$ is a non-negative measure. Substituting $ x=\nu_k(t)$ to the above formula, we obtain the equality
| $ \begin{equation*} f(\nu_k(t))=\int\limits_{\xi\in(-\infty,\infty)}e^{\xi\nu_k(t)}\sigma(d\xi), \ \ \forall\,t\in(a,b),\ \ k=0,1,\ldots,n. \end{equation*}$ |
Hence
| $ \begin{equation*} F(t)=\int\limits_{\xi\in(-\infty,\infty)}g(t,\xi)\,\sigma(d\xi), \ \ \forall\,t\in(a,b). \end{equation*}$ | (5.5) |
Theorem 5.4 is a consequence of Theorem 5.2 and of the properties P1, P2, P4 of the class of exponentially convex functions. $ \square$
Example.
For $ \gamma> 0$, the function $ f(x)=e^{\gamma{}x^2}$ is exponentially
convex on $ (-\infty,\infty)$:
$ e^{\gamma{}x^2}=\int\nolimits_{\xi\in(-\infty,\infty)}e^{\xi{}x}\sigma(d\xi), \ \ \text{where} \ \ \sigma(d\xi)=\frac{1}{2\sqrt{\pi\gamma}}e^{-\xi^2/4\gamma}d\xi.$
Thus the function $ F(t)=\sum\nolimits_{k=0}^ne^{\gamma(\nu_k(t))^2}$ is exponentially convex on $ (-\infty,\infty)$.
Remark 5.4.
Familiarizing himself with our proof of Theorem 5.2, Alexey Kuznetsov (http://www.math.yorku.ca/ãkuznets/) gave a new proof of a somewhat weakened version of this theorem. His proof is based on the theory of stochastic Lévy processes.
References
-
[Akhiezer1965]
(1965) (in Russian). English Transl.: Akhiezer, N.I.: The Clasical Moment Problem. Oliver and Boyd, Edinburgh (1965)
- [Bernstein1928] Bernstein, S.N.: Sur les functions absolument monotones. Acta Math. 52, 1–66 (1928). (in French)
- [Bessis et al.1975] Bessis, D., Moussa, P., Villani, M.: Monotonic converging variational approximations to the functional integrals in quantum statistical mechanics. J. Math. Phys. 16(11), 2318–2325 (1975)
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