Shaohua Fan
School of Science, Tianjin University of Technology and Education, Tianjin, China.
DOI: 10.4236/jamp.2023.114059   PDF    HTML   XML   239 Downloads   881 Views   Citations

Abstract

We study a special class of lower trigonometric matrix value boundary value problems on hyperbolas. Firstly, the pseudo-orthogonal polynomial on hyperbola is given in bilinear form and it is shown that it is the only one. Secondly, a special boundary value problem of lower triangular matrix is presented and transformed into four related boundary value problems. Finally, Liouville theorem and Painlevé theorem and pseudo-orthogonal polynomials are used to give solutions.

Keywords

Hyperbola, Matrix Boundary Value Problem, Orthogonal Polynomial

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1. Introduction

In some references [1] [2] [3] , the boundary value problem (Riemann-Hilbert problem) of analytic functions on finite curves is discussed, but the research on infinite curves is not deep enough. In [4] , the author discusses the Riemann boundary value problem on the positive real axis and generalizes the concept of the generalized principal part.

The Riemann-Hilbert method is a brand-new method for studying orthogonal polynomials formed in recent 20 years. In 1992, FoKas A S, Its A R and Kitaev A V constructed a matrix-valued Riemann-Hilbert boundary value problem in [5] , the only solution of which is the orthogonal polynomial on the real axis. In 1993, Deift P and Zhou X introduced the Riemann-Hilbert boundary value problem of oscillatory type in [6] , and applied it to the study of orthogonal polynomials. Therefore, the Riemann-Hilbert method was formed [6] .

2. Preliminary

In this paper, the right branch of the Hyperbola $x^2-y^2=1$ is denoted by default to L, which is regarded as the image of the function $x=\phi \left(y\right)=\sqrt{y^2+1}$ , and L is oriented from top to bottom.

Denote by $l_a$ the point $\phi \left(a\right)+ia$ and ${\infty }^{\pm }$ respectively its upper and lower infinite ends. Then $ℂ$ consists of two connected components, the right part $S^{+}$ and the left part $S^{-}$ .

We use bilinear form to replace inner product on hyperbola, which is a common way. For example, [Lu J K, 1993] gives the solvable condition of singular integral equation by this way; for example, Delft P. defined a polynomial group similar to orthogonal polynomials in bilinear form in [7] , and we studied similar polynomial groups on hyperbola:

Let $w\left(t\right)$ be a nonzero weight function. We introduce bilinear form in polynomial space ${\Pi }_n$ with degree no more than n:

$\left(f,g\right)={\displaystyle {\int }_{L}w\left(t\right)f\left(t\right)g\left(t\right)\text{d}t},f,g\in {\Pi }_n$ (1)

Take a group of bases $1,t,t^2,\cdots ,t^n$ in ${\Pi }_n$ and make Schmidt orthogonalization on this group of bases, then we have

$p_0\left(z\right)=\frac{1}{{\left({\displaystyle {\int }_{L}w\text{d}t}\right)}^{\frac{1}{2}}}$

$p_1\left(z\right)=\frac{t-\frac{\left(t,p_0\right)}{\left(p_0,p_0\right)}{p}_0}{{\left(t-\frac{\left(t,p_0\right)}{\left(p_0,p_0\right)}{p}_0,t-\frac{\left(t,p_0\right)}{\left(p_0,p_0\right)}{p}_0\right)}^{\frac{1}{2}}}$

$⋮$

$p_n\left(z\right)=\frac{t^n-\frac{\left(t^n,p_{n-1}\right)}{\left(p_{n-1},p_{n-1}\right)}{p}_{n-1}-\cdots -\frac{\left(t^n,p_1\right)}{\left(p_1,p_1\right)}{p}_1-\frac{\left(t^n,p_0\right)}{\left(p_0,p_0\right)}{p}_0}{\left(A_n,A_n\right)}$

where $A_n=t^n-\frac{\left(t^n,p_{n-1}\right)}{\left(p_{n-1},p_{n-1}\right)}{p}_{n-1}-\cdots -\frac{\left(t^n,p_1\right)}{\left(p_1,p_1\right)}{p}_1-\frac{\left(t^n,p_0\right)}{\left(p_0,p_0\right)}{p}_0$ , If $\left(p_n,p_n\right)$ is always not zero, then this process can always be carried out. Finally, we get a pseudo-orthogonal polynomial group with a weight function of $p_0,p_1\left(z\right),\cdots ,p_n\left(z\right)$ on L:

$P_k\left(z\right)=\frac{1}{{\alpha }_k}{p}_k\left(z\right),k=0,1,\cdots ,n,$ (2)

where ${\alpha }_k$ is the first coefficient of $p_k\left(z\right)$ , then $P_k\left(z\right)$ is a pseudo-orthogonal polynomial of degree k with the first coefficient of 1. Obviously, the pseudo-orthogonal polynomial group $P_0,P_1\left(z\right),\cdots ,P_n\left(z\right)$ is unique.

Definition 1. Let f is defined on L, if there is some positive real number a, such that

$\left|f\left(t^{\prime }\right)-f\left(t^{″}\right)\right|\le M{\left|\frac{1}{t^{\prime }}-\frac{1}{t^{″}}\right|}^{\mu },t^{\prime },t^{″}\in \stackrel{⌢}{l_{{\infty }^{+}}{l}_a}\cup \stackrel{⌢}{l_a{l}_{{\infty }^{-}}}$ (3)

where M and $0<\mu \le 1$ are definite constants, then denoted by $f\in {\hat{H}}^{\mu }\left(\infty \right)$ , and if $f\in H^{\mu }\left(L\right)$ , then denoted by $f\in {\hat{H}}^{\mu }\left(L\right)$ . If $f\in {\hat{H}}^{\mu }\left(\infty \right)$ and $f\left(\infty \right)=0$ , then denoted by $f\in {\hat{H}}_0^{\mu }\left(\infty \right)$ , or $f\in {\hat{H}}_0\left(\infty \right)$ . Moreover, if $t^{\lambda }f\in {\hat{H}}_0\left(\infty \right)$ , then denoted by $f\in {\hat{H}}_{\lambda ,0}\left(\infty \right)$ .

Definition 2 Let f is a function defined on L. There exists $t\to \infty$ such that

$f\left(t\right)=\frac{f^{*}\left(t\right)}{t^{\nu }}$ ,

where v is a real number and $f^{*}$ is a bounded function, then denoted by $f\in O^v\left(\infty \right)$ .

Definition 3 If F is holomorphic in the complex plane cut by the Hyperbola, then denoted by $F\in A\left(ℂ\backslash L\right)$ .

Definition 4 Let f be a locally integrable function on L. If

$\left(C\left[f\right]\right)\left(z\right)=\frac{1}{2\pi i}{\displaystyle {\int }_L\frac{f\left(\tau \right)}{\tau -z}\text{d}\tau ,}z\in ℂ\backslash L$ (4)

is integrable, it is called the Cauchy-type integral with kernel density f on L, and the Cauchy principal value integral with kernel density f is defined by

$\left(C\left[f\right]\right)\left(t\right)=\frac{1}{2\pi i}{\displaystyle {\int }_L\frac{f\left(\tau \right)}{\tau -t}\text{d}\tau }=\underset{r\to 0^{+}}{\lim }\frac{1}{2\pi i}{\displaystyle {\int }_{\left|y-a\right|>0}\frac{f\left(\left(\phi \left(y\right)+iy\right)\right)\left({\phi }^{\prime }\left(y\right)+iy\right)}{\phi \left(y\right)+iy-t}\text{d}y}$ (5)

where $t=\phi \left(a\right)+ia\in L$ , if the integral exists.

Ref [Wang Ying, 2017], below we introduce the concept of a generalized main part.

Definition 5 Let $F\in A\left(ℂ\backslash L\right)$ . If there exists an entire function $E\left(z\right)$ such that

$\underset{z\to \infty }{\lim }\left[F\left(z\right)-E\left(z\right)\right]=0$ . (6)

and then $E\left(z\right)$ is called the generalized principal part of $F\left(z\right)$ at $\infty$ , denoted by $\text{G}\text{.P}\left[F,\infty \right]$ .

Reference [8] proves the generalized principal part of Cauchy integral at infinity and Plemelj formula.

Theorem 1 [8] If $f\in H\left(L\right)\cap O^v\left(\infty \right)\left(v>0\right)$ is locally integrable on L. Then

$\text{G}\text{.P}\left[C\left[f\right],\infty \right]=0$ . (7)

Theorem 2 [8] If $f\in {\hat{H}}^{\mu }\left(L\right)$ , then the boundary values of the Cauchy-type integral $C\left[f\right]$ exist and have the following Plemelj formula:

${\left(C\left[f\right]\right)}^{\pm }\left(t\right)=\pm \frac{1}{2}f\left(t\right)+\frac{1}{2\pi i}{\displaystyle {\int }_L\frac{f\left(\tau \right)}{\tau -t}\text{d}\tau }$ . (8)

3. Matrix Value Riemann Boundary Value Problem

In this paper, we consider the Riemann boundary value problem of lower trigonometric matrix on hyperbola.

Let

$\Phi \left(z\right)=\left(\begin{array}{cc}{\Phi }_{1,1}\left(z\right)& {\Phi }_{1,2}\left(z\right)\\ {\Phi }_{2,1}\left(z\right)& {\Phi }_{2,2}\left(z\right)\end{array}\right)$ (9)

be a matrix-valued function defined on subset $\Omega$ of the complex plane $ℂ$ , and each element ${\Phi }_{j,k}$ be a function defined on $\Omega$ . If every element ${\Phi }_{j,k}$ of $\Phi$ satisfies the same property, then $\Phi$ is said to have its corresponding property, such as $\Phi \in A\left(ℂ\backslash L\right),\text{G}\text{.P}\left[\Phi ,\infty \right]\left(z\right),\Phi \in H\left(L\right)$ .

Problem (boundary value problem of lower trigonometric matrix value function) Find the matrix-valued partitioned holomorphic function $\Phi$ with L as the jump curve, such that

$\{\begin{array}{l}{\Phi }^{+}\left(t\right)=\left(\begin{array}{cc}1& 0\\ w\left(t\right)& 1\end{array}\right){\Phi }^{-}\left(t\right),t\in L,\\ \text{G}\text{.P}\left[\Xi \Phi ,\infty \right]\left(z\right)=I,\end{array}$ (10)

where

$\Xi \left(z\right)=\left(\begin{array}{cc}{z}^{-n}& 0\\ 0& z^n\end{array}\right),$ (11)

I is the identity matrix of 2 × 2, $w\in H^{\mu }\left(L\right)\cap {\hat{H}}_{2n,0}\left(\infty \right)$ .

We can convert (10) into four related Riemann boundary value problems:

$\{\begin{array}{l}{\Phi }_{1,1}^{+}\left(t\right)={\Phi }_{1,1}^{-}\left(t\right),t\in L,\\ \text{G}\text{.P}\left[z^{-n}{\Phi }_{1,1},\infty \right]=1,\end{array}$ (12)

$\{\begin{array}{l}{\Phi }_{1,2}^{+}\left(t\right)={\Phi }_{1,2}^{-}\left(t\right),t\in L,\\ \text{G}\text{.P}\left[z^{-n}{\Phi }_{1,2},\infty \right]=0,\end{array}$ (13)

$\{\begin{array}{l}{\Phi }_{2,1}^{+}\left(t\right)={\Phi }_{2,1}^{-}\left(t\right)+w\left(t\right){\Phi }_{1,1}^{-}\left(t\right),t\in L,\\ \text{G}\text{.P}\left[z^n{\Phi }_{2,1},\infty \right]=0,\end{array}$ (14)

$\{\begin{array}{l}{\Phi }_{2,2}^{+}\left(t\right)={\Phi }_{2,2}^{-}\left(t\right)+w\left(t\right){\Phi }_{1,2}^{-}\left(t\right),t\in L,\\ \text{G}\text{.P}\left[z^n{\Phi }_{2,2},\infty \right]=1.\end{array}$ (15)

Obviously, (12) is a Liouville problem. It is known from Painlevé theorem that ${\Phi }_{1,1}\left(z\right)$ is analytic over the entire complex plane. Because $\text{G}\text{.P}\left[z^{-n}{\Phi }_{1,1},\infty \right]=1$ , it is known from the generalized Liouville theorem that

${\Phi }_{1,1}\left(z\right)=P_n\left(z\right),$ (16)

where $P_n\left(z\right)$ is a polynomial with a leading coefficient of 1 and a degree of n.

By (16), we have

$\{\begin{array}{l}{\Phi }_{2,1}^{+}\left(t\right)={\Phi }_{2,1}^{-}\left(t\right)+w\left(t\right)P_n\left(t\right),t\in L,\\ \text{G}\text{.P}\left[z^n{\Phi }_{2,1},\infty \right]=0,\end{array}$ (17)

Obviously (17) is a jump problem with L as the jump curve. Let

$\psi \left(z\right)=C\left[w{P}_n\right]\left(z\right)=\frac{1}{2\pi i}{\displaystyle {\int }_L\frac{w\left(\tau \right)P_n\left(\tau \right)}{\tau -z}\text{d}\tau },z\in L$ , (18)

by $w\in H^{\mu }\left(L\right)\cap {\hat{H}}_{2n,0}\left(\infty \right)$ ,

$w{P}_n\in H^{\mu }\left(L\right)\cap {\hat{H}}_{n,0}\left(\infty \right)$ . (19)

Therefore, by Plemelj formula (8) and Theorem 1, we can know that $\psi \left(z\right)$ is a partitioned holomorphic function with L as the jump curve, and satisfies:

$\{\begin{array}{l}{\psi }^{+}\left(t\right)={\psi }^{-}\left(t\right)+\omega \left(t\right)P_n\left(t\right),t\in L,\\ \text{G}\text{.P}\left[\psi ,\infty \right]\left(z\right)=0,\end{array}$ (20)

let $F\left(z\right)=\Phi \left(z\right)-\psi \left(z\right)$ , then F is a partitioned holomorphic function with L as the jump curve and satisfies:

$\{\begin{array}{l}{F}^{+}\left(t\right)=F^{-}\left(t\right),t\in L,\\ \text{G}\text{.P}\left[F,\infty \right]=\text{0,}\end{array}$ (21)

Obviously problem (21) is a zero-order Liouville problem, its solution is $F\left(z\right)=0$ , so

${\Phi }_{2,1}\left(z\right)=C\left[w{P}_n\right]\left(z\right)=\frac{1}{2\pi i}{\displaystyle {\int }_L\frac{w\left(\tau \right)P_n\left(\tau \right)}{\tau -z}\text{d}\tau },z\in ℂ\backslash L$ (22)

if and only if condition $\text{G}\text{.P}\left[z^n{\Phi }_{2,1},\infty \right]=0$ is satisfied. By

$\begin{array}{c}{z}^n{\Phi }_{2,1}\left(z\right)=\frac{1}{2\pi i}{\displaystyle {\int }_L\frac{w\left(\tau \right)P_n\left(\tau \right)\left(z^n-{\tau }^n\right)}{\tau -z}\text{d}\tau }+\frac{1}{2\pi i}{\displaystyle {\int }_L\frac{w\left(\tau \right)P_n\left(\tau \right){\tau }^n}{\tau -z}\text{d}\tau }\\ =-{\displaystyle \underset{k=0}{\overset{n-1}{\sum }}\frac{z^k}{2\pi i}}{\displaystyle {\int }_{L}w\left(\tau \right)P_n\left(\tau \right){\tau }^{n-1-k}\text{d}\tau }+\frac{1}{2\pi i}{\displaystyle {\int }_L\frac{w\left(\tau \right)P_n\left(\tau \right){\tau }^n}{\tau -z}\text{d}\tau },\end{array}$ (23)

and Theorem 1 and (19), it can be seen that $\text{G}\text{.P}\left[z^n{\Phi }_{2,1},\infty \right]=0$ is equivalent to

$\frac{1}{2\pi i}{\displaystyle {\int }_{L}w\left(\tau \right)P_n\left(\tau \right){\tau }^k\text{d}\tau }=0,k=0,1,\cdots ,n-1$ . (24)

Obviously (13) is the Liouville problem, similar to (12) we have

${\Phi }_{1,2}\left(z\right)=q_{n-1}\left(z\right)$ (25)

where $q_{n-1}\left(z\right)$ is a polynomial of order not exceeding $n-1$ .

By (16), we have

$\{\begin{array}{l}{\Phi }_{2,2}^{+}\left(t\right)={\Phi }_{2,2}^{-}\left(t\right)+w\left(t\right)q_{n-1}\left(t\right),t\in L,\\ \text{G}\text{.P}\left[z^n{\Phi }_{2,2},\infty \right]=1.\end{array}$ (26)

Obviously, (26) is a fixed-order jump problem, similar to (15). It can be seen that its solution is

${\Phi }_{2,2}\left(z\right)=C\left[w{q}_{n-1}\right]\left(z\right)=\frac{1}{2\pi i}{\displaystyle {\int }_L\frac{w\left(\tau \right)q_{n-1}\left(\tau \right)}{\tau -z}\text{d}\tau },z\in ℂ\backslash L$ (27)

if and only if condition

$\{\begin{array}{l}\frac{1}{2\pi i}{\displaystyle {\int }_{L}w\left(\tau \right)q_{n-1}\left(\tau \right){\tau }^k\text{d}\tau }=0,k=0,1,\cdots ,n-2,\\ \frac{1}{2\pi i}{\displaystyle {\int }_{L}w\left(\tau \right)q_{n-1}\left(\tau \right){\tau }^{n-1}\text{d}\tau }=-1,\end{array}$ (28)

is satisfied.

Let $q_{n-1}=\lambda P_{n-1}$ , then

$\frac{1}{2\pi i}{\displaystyle {\int }_L\omega \left(\tau \right)\lambda P_{n-1}{P}_{n-1}}=-1,$ (29)

that is,

$\lambda =\frac{-2\pi i}{{\displaystyle {\int }_L\omega \left(\tau \right)P_{n-1}^2\left(\tau \right)\text{d}\tau }}$ (30)

then $q_{n-1}$ is a pseudo-orthogonal polynomial of degree $n-1$ on L with respect to the weight function w.

Definition 6

$f^{*}\left(z\right)=\frac{1}{2\pi i}{\displaystyle {\int }_L\frac{\omega \left(\tau \right)f\left(\tau \right)}{\tau -z}\text{d}\tau },z\notin L,$ (31)

we call it the companion function of f with respect to the weight function w.

Theorem 3 If $w\in H^{\mu }\left(L\right)\cap {\hat{H}}_{2n,0}\left(\infty \right)$ , then the lower triangular matrix-valued Riemann boundary value problem (10) has a solution, and its solution has the following form:

$\Phi \left(z\right)=\left(\begin{array}{cc}{P}_n\left(z\right)& \lambda P_{n-1}\left(z\right)\\ P_n^{*}\left(z\right)& \lambda P_{n-1}^{*}\left(z\right)\end{array}\right),$ (32)

where $P_n\left(z\right)$ is a polynomial with a leading coefficient of 1 and a degree of n, and $P_n^{*}$ is the companion function of $P_n$ with respect to the middle weight function w.

Proof: If (10) has a solution, it can be seen from the previous discussion that its solution is of the form (32).

Conversely, the polynomial with pseudo-orthogonal and leading coefficient 1 is unique, and by reversing each previous step, we get that $\Phi$ is the solution of (10), that is, (10) has and only one set of solutions (32).

The matrix-valued boundary value problem (10) is characterized by the pseudo-orthogonal polynomial $P_n$ on L with respect to the weight function w and the leading coefficient is 1. Therefore, we call this problem the Riemann-Hilbert characteristic characterization of the orthogonal polynomial of the weight function w on hyperbola, or $P_n$ is the characteristic orthogonal polynomial of the matrix-valued boundary value problem (10), please refer to [Deift P, 2011] for details.

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.

References

[1]	Gakhov, F.D. (1977) Boundary Value Problems. Nauka, Moscow.
[2]	Lu, J.K. (1993) Boundary Value Problems for Analytic Functions. World Scientific, Singapore. [CrossRef]
[3]	Muskhelishvili, N.I. (1953) Singular Integral Equations. 2nd Edition, P. Noordhoff N. V., Groningen.
[4]	Wang, Y., Duan, P. and Du, J.Y. (2017) Riemann Boundary Value Problems on Positive Real Axis. Science in China: Mathematics, 47, 887-918. [CrossRef]
[5]	Fokas, A.S., Its, A. and Kitaev, A.V. (1992) The Isomonodromy Approach to Matric Models in 2D Quantum Gravity. Communications in Mathematical Physics, 147, 395-430. [CrossRef]
[6]	Deift, P. and Zhou, X. (1993) A Steepest Descent Method for Oscillatory Riemann—Hilbert Problems. Asymptotics for the MKdV Equation. Annals of Mathematics, 137, 295-368. [CrossRef]
[7]	Deift, P., Its, A. and Krasovsky, I. (2011) Asymptotics of Toeplitz, Hankel, and Toeplitz + Hankel Determinants with Fisher-Hartwig Singularities. Annals of Mathematics, 174, 1243-1299. [CrossRef]
[8]	Wei, Y.Q. and Liu, H. (2023) Properties of Cauchy Integral on the Hyperbola. Journal of Ningxia Normal University, 44, 6-12.