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.
In some references [
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 [
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 = φ ( y ) = y 2 + 1 , and L is oriented from top to bottom.
Denote by l a the point φ ( a ) + i a and ∞ ± 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 [
Let w ( t ) be a nonzero weight function. We introduce bilinear form in polynomial space Π n with degree no more than n:
( f , g ) = ∫ L w ( t ) f ( t ) g ( t ) d t , f , g ∈ Π n (1)
Take a group of bases 1 , t , t 2 , ⋯ , t n in Π n and make Schmidt orthogonalization on this group of bases, then we have
p 0 ( z ) = 1 ( ∫ L w d t ) 1 2
p 1 ( z ) = t − ( t , p 0 ) ( p 0 , p 0 ) p 0 ( t − ( t , p 0 ) ( p 0 , p 0 ) p 0 , t − ( t , p 0 ) ( p 0 , p 0 ) p 0 ) 1 2
⋮
p n ( z ) = t n − ( t n , p n − 1 ) ( p n − 1 , p n − 1 ) p n − 1 − ⋯ − ( t n , p 1 ) ( p 1 , p 1 ) p 1 − ( t n , p 0 ) ( p 0 , p 0 ) p 0 ( A n , A n )
where A n = t n − ( t n , p n − 1 ) ( p n − 1 , p n − 1 ) p n − 1 − ⋯ − ( t n , p 1 ) ( p 1 , p 1 ) p 1 − ( t n , p 0 ) ( p 0 , p 0 ) p 0 , If ( p n , p n ) 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 ( z ) , ⋯ , p n ( z ) on L:
P k ( z ) = 1 α k p k ( z ) , k = 0 , 1 , ⋯ , n , (2)
where α k is the first coefficient of p k ( z ) , then P k ( z ) is a pseudo-orthogonal polynomial of degree k with the first coefficient of 1. Obviously, the pseudo-orthogonal polynomial group P 0 , P 1 ( z ) , ⋯ , P n ( z ) is unique.
Definition 1. Let f is defined on L, if there is some positive real number a, such that
| f ( t ′ ) − f ( t ″ ) | ≤ M | 1 t ′ − 1 t ″ | μ , t ′ , t ″ ∈ l ∞ + l a ⌢ ∪ l a l ∞ − ⌢ (3)
where M and 0 < μ ≤ 1 are definite constants, then denoted by f ∈ H ^ μ ( ∞ ) , and if f ∈ H μ ( L ) , then denoted by f ∈ H ^ μ ( L ) . If f ∈ H ^ μ ( ∞ ) and f ( ∞ ) = 0 , then denoted by f ∈ H ^ 0 μ ( ∞ ) , or f ∈ H ^ 0 ( ∞ ) . Moreover, if t λ f ∈ H ^ 0 ( ∞ ) , then denoted by f ∈ H ^ λ , 0 ( ∞ ) .
Definition 2 Let f is a function defined on L. There exists t → ∞ such that
f ( t ) = f * ( t ) t ν ,
where v is a real number and f * is a bounded function, then denoted by f ∈ O v ( ∞ ) .
Definition 3 If F is holomorphic in the complex plane cut by the Hyperbola, then denoted by F ∈ A ( ℂ \ L ) .
Definition 4 Let f be a locally integrable function on L. If
( C [ f ] ) ( z ) = 1 2 π i ∫ L f ( τ ) τ − z d τ , z ∈ ℂ \ 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
( C [ f ] ) ( t ) = 1 2 π i ∫ L f ( τ ) τ − t d τ = lim r → 0 + 1 2 π i ∫ | y − a | > 0 f ( ( φ ( y ) + i y ) ) ( φ ′ ( y ) + i y ) φ ( y ) + i y − t d y (5)
where t = φ ( a ) + i a ∈ L , if the integral exists.
Ref [Wang Ying, 2017], below we introduce the concept of a generalized main part.
Definition 5 Let F ∈ A ( ℂ \ L ) . If there exists an entire function E ( z ) such that
lim z → ∞ [ F ( z ) − E ( z ) ] = 0 . (6)
and then E ( z ) is called the generalized principal part of F ( z ) at ∞ , denoted by G .P [ F , ∞ ] .
Reference [
Theorem 1 [
G .P [ C [ f ] , ∞ ] = 0 . (7)
Theorem 2 [
( C [ f ] ) ± ( t ) = ± 1 2 f ( t ) + 1 2 π i ∫ L f ( τ ) τ − t d τ . (8)
In this paper, we consider the Riemann boundary value problem of lower trigonometric matrix on hyperbola.
Let
Φ ( z ) = ( Φ 1 , 1 ( z ) Φ 1 , 2 ( z ) Φ 2 , 1 ( z ) Φ 2 , 2 ( z ) ) (9)
be a matrix-valued function defined on subset Ω of the complex plane ℂ , and each element Φ j , k be a function defined on Ω . If every element Φ j , k of Φ satisfies the same property, then Φ is said to have its corresponding property, such as Φ ∈ A ( ℂ \ L ) , G .P [ Φ , ∞ ] ( z ) , Φ ∈ H ( L ) .
Problem (boundary value problem of lower trigonometric matrix value function) Find the matrix-valued partitioned holomorphic function Φ with L as the jump curve, such that
{ Φ + ( t ) = ( 1 0 w ( t ) 1 ) Φ − ( t ) , t ∈ L , G .P [ Ξ Φ , ∞ ] ( z ) = I , (10)
where
Ξ ( z ) = ( z − n 0 0 z n ) , (11)
I is the identity matrix of 2 × 2, w ∈ H μ ( L ) ∩ H ^ 2 n , 0 ( ∞ ) .
We can convert (10) into four related Riemann boundary value problems:
{ Φ 1 , 1 + ( t ) = Φ 1 , 1 − ( t ) , t ∈ L , G .P [ z − n Φ 1 , 1 , ∞ ] = 1 , (12)
{ Φ 1 , 2 + ( t ) = Φ 1 , 2 − ( t ) , t ∈ L , G .P [ z − n Φ 1 , 2 , ∞ ] = 0 , (13)
{ Φ 2 , 1 + ( t ) = Φ 2 , 1 − ( t ) + w ( t ) Φ 1 , 1 − ( t ) , t ∈ L , G .P [ z n Φ 2 , 1 , ∞ ] = 0 , (14)
{ Φ 2 , 2 + ( t ) = Φ 2 , 2 − ( t ) + w ( t ) Φ 1 , 2 − ( t ) , t ∈ L , G .P [ z n Φ 2 , 2 , ∞ ] = 1. (15)
Obviously, (12) is a Liouville problem. It is known from Painlevé theorem that Φ 1 , 1 ( z ) is analytic over the entire complex plane. Because G .P [ z − n Φ 1 , 1 , ∞ ] = 1 , it is known from the generalized Liouville theorem that
Φ 1 , 1 ( z ) = P n ( z ) , (16)
where P n ( z ) is a polynomial with a leading coefficient of 1 and a degree of n.
By (16), we have
{ Φ 2 , 1 + ( t ) = Φ 2 , 1 − ( t ) + w ( t ) P n ( t ) , t ∈ L , G .P [ z n Φ 2 , 1 , ∞ ] = 0 , (17)
Obviously (17) is a jump problem with L as the jump curve. Let
ψ ( z ) = C [ w P n ] ( z ) = 1 2 π i ∫ L w ( τ ) P n ( τ ) τ − z d τ , z ∈ L , (18)
by w ∈ H μ ( L ) ∩ H ^ 2 n , 0 ( ∞ ) ,
w P n ∈ H μ ( L ) ∩ H ^ n , 0 ( ∞ ) . (19)
Therefore, by Plemelj formula (8) and Theorem 1, we can know that ψ ( z ) is a partitioned holomorphic function with L as the jump curve, and satisfies:
{ ψ + ( t ) = ψ − ( t ) + ω ( t ) P n ( t ) , t ∈ L , G .P [ ψ , ∞ ] ( z ) = 0 , (20)
let F ( z ) = Φ ( z ) − ψ ( z ) , then F is a partitioned holomorphic function with L as the jump curve and satisfies:
{ F + ( t ) = F − ( t ) , t ∈ L , G .P [ F , ∞ ] = 0, (21)
Obviously problem (21) is a zero-order Liouville problem, its solution is F ( z ) = 0 , so
Φ 2 , 1 ( z ) = C [ w P n ] ( z ) = 1 2 π i ∫ L w ( τ ) P n ( τ ) τ − z d τ , z ∈ ℂ \ L (22)
if and only if condition G .P [ z n Φ 2 , 1 , ∞ ] = 0 is satisfied. By
z n Φ 2 , 1 ( z ) = 1 2 π i ∫ L w ( τ ) P n ( τ ) ( z n − τ n ) τ − z d τ + 1 2 π i ∫ L w ( τ ) P n ( τ ) τ n τ − z d τ = − ∑ k = 0 n − 1 z k 2 π i ∫ L w ( τ ) P n ( τ ) τ n − 1 − k d τ + 1 2 π i ∫ L w ( τ ) P n ( τ ) τ n τ − z d τ , (23)
and Theorem 1 and (19), it can be seen that G .P [ z n Φ 2 , 1 , ∞ ] = 0 is equivalent to
1 2 π i ∫ L w ( τ ) P n ( τ ) τ k d τ = 0 , k = 0 , 1 , ⋯ , n − 1 . (24)
Obviously (13) is the Liouville problem, similar to (12) we have
Φ 1 , 2 ( z ) = q n − 1 ( z ) (25)
where q n − 1 ( z ) is a polynomial of order not exceeding n − 1 .
By (16), we have
{ Φ 2 , 2 + ( t ) = Φ 2 , 2 − ( t ) + w ( t ) q n − 1 ( t ) , t ∈ L , G .P [ z n Φ 2 , 2 , ∞ ] = 1. (26)
Obviously, (26) is a fixed-order jump problem, similar to (15). It can be seen that its solution is
Φ 2 , 2 ( z ) = C [ w q n − 1 ] ( z ) = 1 2 π i ∫ L w ( τ ) q n − 1 ( τ ) τ − z d τ , z ∈ ℂ \ L (27)
if and only if condition
{ 1 2 π i ∫ L w ( τ ) q n − 1 ( τ ) τ k d τ = 0 , k = 0 , 1 , ⋯ , n − 2 , 1 2 π i ∫ L w ( τ ) q n − 1 ( τ ) τ n − 1 d τ = − 1 , (28)
is satisfied.
Let q n − 1 = λ P n − 1 , then
1 2 π i ∫ L ω ( τ ) λ P n − 1 P n − 1 = − 1 , (29)
that is,
λ = − 2 π i ∫ L ω ( τ ) P n − 1 2 ( τ ) d τ (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 * ( z ) = 1 2 π i ∫ L ω ( τ ) f ( τ ) τ − z d τ , z ∉ L , (31)
we call it the companion function of f with respect to the weight function w.
Theorem 3 If w ∈ H μ ( L ) ∩ H ^ 2 n , 0 ( ∞ ) , then the lower triangular matrix-valued Riemann boundary value problem (10) has a solution, and its solution has the following form:
Φ ( z ) = ( P n ( z ) λ P n − 1 ( z ) P n * ( z ) λ P n − 1 * ( z ) ) , (32)
where P n ( z ) 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 Φ 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.
The author declares no conflicts of interest regarding the publication of this paper.
Fan, S.H. (2023) Matrix Boundary Value Problem on Hyperbola. Journal of Applied Mathematics and Physics, 11, 884-890. https://doi.org/10.4236/jamp.2023.114059