Misplaced Pages

#164835

49-433: Suppose that Σ {\displaystyle \Sigma } is a smooth , simple, closed contour in the complex z {\displaystyle z} plane . Divide the plane into two parts denoted by Σ + {\displaystyle \Sigma _{+}} (the inside) and Σ − {\displaystyle \Sigma _{-}} (the outside), determined by

98-419: A constraint on the blow-up) of M {\displaystyle M} near the special points 1 {\displaystyle 1} and − 1 {\displaystyle -1} is crucial. Otherwise any function of the form is also a solution. In general, conditions on growth are necessary at special points (the end-points of the jump contour or intersection point) to ensure that

147-806: A function of r and θ , or as a family of functions of θ indexed by r . If D = { z : | z | < 1 } {\displaystyle D=\{z:|z|<1\}} is the open unit disc in C , T is the boundary of the disc, and f a function on T that lies in L ( T ), then the function u given by u ( r e i θ ) = 1 2 π ∫ − π π P r ( θ − t ) f ( e i t ) d t , 0 ≤ r < 1 {\displaystyle u(re^{i\theta })={\frac {1}{2\pi }}\int _{-\pi }^{\pi }P_{r}(\theta -t)f(e^{it})\,\mathrm {d} t,\quad 0\leq r<1}

196-471: Is harmonic in D and has a radial limit that agrees with f almost everywhere on the boundary T of the disc. That the boundary value of u is f can be argued using the fact that as r → 1 , the functions P r ( θ ) form an approximate unit in the convolution algebra L ( T ). As linear operators, they tend to the Dirac delta function pointwise on L ( T ). By the maximum principle , u

245-865: Is a circle, the problem reduces to deriving the Poisson formula . By the Riemann mapping theorem , it suffices to consider the case when Σ {\displaystyle \Sigma } is the circle group T = { z ∈ C : | z | = 1 } {\textstyle \mathbb {T} =\{z\in \mathbb {C} :|z|=1\}} . In this case, one may seek M + ( z ) {\displaystyle M_{+}(z)} along with its Schwarz reflection For z ∈ T {\displaystyle z\in \mathbb {T} } , one has z = 1 / z ¯ {\displaystyle z=1/{\bar {z}}} and so Hence

294-612: Is based on the analysis of d-bar problems, rather than the asymptotic analysis of singular integrals on contours. An alternative way of dealing with jump matrices with no analytic extensions was introduced in Varzugin (1996) . Another extension of the theory appears in Kamvissis & Teschl (2012) where the underlying space of the Riemann–Hilbert problem is a compact hyperelliptic Riemann surface . The correct factorization problem

343-575: Is bounded, what is the solution M {\displaystyle M} ? To solve this, let's take the logarithm of equation M + = G M − {\displaystyle M_{+}=GM_{-}} . Since M ( z ) {\displaystyle M(z)} tends to 1 {\displaystyle 1} , log ⁡ M → 0 {\displaystyle \log M\to 0} as z → ∞ {\displaystyle z\to \infty } . A standard fact about

392-480: Is clear from the properties of the Fourier transform that, at least formally, the convolution P [ u ] ( t , x ) = [ P ( t , ⋅ ) ∗ u ] ( x ) {\displaystyle P[u](t,x)=[P(t,\cdot )*u](x)} is a solution of Laplace's equation in the upper half-plane. One can also show that as t → 0 , P [ u ]( t , x ) → u ( x ) in

441-474: Is given by P y ( x ) = 1 π y x 2 + y 2 . {\displaystyle P_{y}(x)={\frac {1}{\pi }}{\frac {y}{x^{2}+y^{2}}}.} Given a function f ∈ L p ( R ) {\displaystyle f\in L^{p}(\mathbb {R} )} , the L space of integrable functions on

490-732: Is harmonic on the ball B r {\displaystyle B_{r}} and that P [ u ]( x ) extends to a continuous function on the closed ball of radius r , and the boundary function coincides with the original function  u . An expression for the Poisson kernel of an upper half-space can also be obtained. Denote the standard Cartesian coordinates of R n + 1 {\displaystyle \mathbb {R} ^{n+1}} by ( t , x ) = ( t , x 1 , … , x n ) . {\displaystyle (t,x)=(t,x_{1},\dots ,x_{n}).} The upper half-space

539-456: Is no more holomorphic, but rather meromorphic , by reason of the Riemann–Roch theorem . The related singular kernel is not the usual Cauchy kernel, but rather a more general kernel involving meromorphic differentials defined naturally on the surface (see e.g. the appendix in Kamvissis & Teschl (2012) ). The Riemann–Hilbert problem deformation theory is applied to the problem of stability of

SECTION 10

#1732780324165

588-1170: Is replaced by in a region R {\displaystyle R} , where M ¯ {\displaystyle {\overline {M}}} is the complex conjugate of M {\displaystyle M} and A ( z , z ¯ ) {\displaystyle A(z,{\bar {z}})} and B ( z , z ¯ ) {\displaystyle B(z,{\bar {z}})} are functions of z {\displaystyle z} and z ¯ {\displaystyle {\bar {z}}} . Generalized analytic functions have applications in differential geometry , in solving certain type of multidimensional nonlinear partial differential equations and multidimensional inverse scattering . Riemann–Hilbert problems have applications to several related classes of problems. The numerical analysis of Riemann–Hilbert problems can provide an effective way for numerically solving integrable PDEs (see e.g. Trogdon & Olver (2016) ). In particular, Riemann–Hilbert factorization problems are used to extract asymptotic values for

637-510: Is the surface area of the unit ( n  − 1)-sphere . Then, if u ( x ) is a continuous function defined on S , the corresponding Poisson integral is the function P [ u ]( x ) defined by P [ u ] ( x ) = ∫ S u ( ζ ) P ( x , ζ ) d σ ( ζ ) . {\displaystyle P[u](x)=\int _{S}u(\zeta )P(x,\zeta )\,d\sigma (\zeta ).} It can be shown that P [ u ]( x )

686-466: Is the boundary value of g + h , where g (resp. h ) is a holomorphic (resp. antiholomorphic ) function on D . When one also asks for the harmonic extension to be holomorphic, then the solutions are elements of a Hardy space . This is true when the negative Fourier coefficients of f all vanish. In particular, the Poisson kernel is commonly used to demonstrate the equivalence of the Hardy spaces on

735-557: Is the complex form of the nonhomogeneous Cauchy-Riemann equations . To show this, let with u ( x , y ) {\displaystyle u(x,y)} , v ( x , y ) {\displaystyle v(x,y)} , g ( x , y ) {\displaystyle g(x,y)} and h ( x , y ) {\displaystyle h(x,y)} all real-valued functions of real variables x {\displaystyle x} and y {\displaystyle y} . Then, using

784-471: Is the only such harmonic function on D . Convolutions with this approximate unit gives an example of a summability kernel for the Fourier series of a function in L ( T ) ( Katznelson 1976 ). Let f ∈ L ( T ) have Fourier series { f k }. After the Fourier transform , convolution with P r ( θ ) becomes multiplication by the sequence { r } ∈ ℓ ( Z ). Taking the inverse Fourier transform of

833-913: Is the set defined by H n + 1 = { ( t ; x ) ∈ R n + 1 ∣ t > 0 } . {\displaystyle H^{n+1}=\left\{(t;x)\in \mathbb {R} ^{n+1}\mid t>0\right\}.} The Poisson kernel for H is given by P ( t , x ) = c n t ( t 2 + ‖ x ‖ 2 ) ( n + 1 ) / 2 {\displaystyle P(t,x)=c_{n}{\frac {t}{\left(t^{2}+\left\|x\right\|^{2}\right)^{(n+1)/2}}}} where c n = Γ [ ( n + 1 ) / 2 ] π ( n + 1 ) / 2 . {\displaystyle c_{n}={\frac {\Gamma [(n+1)/2]}{\pi ^{(n+1)/2}}}.} The Poisson kernel for

882-890: Is to find a pair of analytic functions M + ( t ) {\displaystyle M_{+}(t)} and M − ( t ) {\displaystyle M_{-}(t)} on the "+" and "−" side of Σ {\displaystyle \Sigma } , respectively, such that for t ∈ Σ {\displaystyle t\in \Sigma } one has where α ( t ) {\displaystyle \alpha (t)} , β ( t ) {\displaystyle \beta (t)} and γ ( t ) {\displaystyle \gamma (t)} are given complex-valued functions. Given an oriented contour Σ {\displaystyle \Sigma } (technically: an oriented union of smooth curves without points of infinite self-intersection in

931-616: The L 2 {\displaystyle L^{2}} -sense . At end-points or intersection points of the contour Σ {\displaystyle \Sigma } , the jump condition is not defined; constraints on the growth of M {\displaystyle M} near those points have to be posed to ensure uniqueness (see the scalar problem below). Suppose G = 2 {\displaystyle G=2} and Σ = [ − 1 , 1 ] {\displaystyle \Sigma =[-1,1]} . Assuming M {\displaystyle M}

980-637: The Cauchy transform is that C + − C − = I {\displaystyle C_{+}-C_{-}=I} where C + {\displaystyle C_{+}} and C − {\displaystyle C_{-}} are the limits of the Cauchy transform from above and below Σ {\displaystyle \Sigma } ; therefore, we get when z ∈ Σ {\displaystyle z\in \Sigma } . Because

1029-447: The Lax pair ) is not self-adjoint , by Kamvissis, McLaughlin & Miller (2003) . In that case, actual "steepest descent contours" are defined and computed. The corresponding variational problem is a max-min problem: one looks for a contour that minimizes the "equilibrium" measure. The study of the variational problem and the proof of existence of a regular solution, under some conditions on

SECTION 20

#1732780324165

1078-468: The index of the contour with respect to a point. The classical problem, considered in Riemann's PhD dissertation, was that of finding a function analytic inside Σ + {\displaystyle \Sigma _{+}} , such that the boundary values of M + {\displaystyle M_{+}} along Σ {\displaystyle \Sigma } satisfy

1127-516: The unit disk . The kernel can be understood as the derivative of the Green's function for the Laplace equation. It is named for Siméon Poisson . Poisson kernels commonly find applications in control theory and two-dimensional problems in electrostatics . In practice, the definition of Poisson kernels are often extended to n -dimensional problems. In the complex plane , the Poisson kernel for

1176-632: The DBAR problem yields As such, if M {\displaystyle M} is holomorphic for z ∈ D {\displaystyle z\in D} , then the Cauchy-Riemann equations must be satisfied. In case M → 1 {\displaystyle M\to 1} as z → ∞ {\displaystyle z\to \infty } and D := C {\displaystyle D:=\mathbb {C} } ,

1225-488: The Deift–Zhou analysis is the asymptotic analysis of singular integrals on contours. The relevant kernel is the standard Cauchy kernel (see Gakhov (2001) ; also cf. the scalar example below). An essential extension of the nonlinear method of stationary phase has been the introduction of the so-called finite gap g-function transformation by Deift, Venakides & Zhou (1997) , which has been crucial in most applications. This

1274-461: The Hardy space H on the upper half-plane is a Banach space , and, in particular, its restriction to the real axis is a closed subspace of L p ( R ) . {\displaystyle L^{p}(\mathbb {R} ).} The situation is only analogous to the case for the unit disk; the Lebesgue measure for the unit circle is finite, whereas that for the real line is not. For

1323-481: The Riemann problem as well as Hilbert's generalization, the contour Σ {\displaystyle \Sigma } was simple. A full Riemann–Hilbert problem allows that the contour may be composed of a union of several oriented smooth curves, with no intersections. The "+" and "−" sides of the "contour" may then be determined according to the index of a point with respect to Σ {\displaystyle \Sigma } . The Riemann–Hilbert problem

1372-552: The Wikimedia System Administrators, please include the details below. Request from 172.68.168.226 via cp1108 cp1108, Varnish XID 216091666 Upstream caches: cp1108 int Error: 429, Too Many Requests at Thu, 28 Nov 2024 07:52:04 GMT Poisson kernel In mathematics, and specifically in potential theory , the Poisson kernel is an integral kernel , used for solving the two-dimensional Laplace equation , given Dirichlet boundary conditions on

1421-945: The ball of radius r , B r ⊂ R n , {\displaystyle r,B_{r}\subset \mathbb {R} ^{n},} the Poisson kernel takes the form P ( x , ζ ) = r 2 − | x | 2 r ω n − 1 | x − ζ | n {\displaystyle P(x,\zeta )={\frac {r^{2}-|x|^{2}}{r\omega _{n-1}|x-\zeta |^{n}}}} where x ∈ B r , ζ ∈ S {\displaystyle x\in B_{r},\zeta \in S} (the surface of B r {\displaystyle B_{r}} ), and ω n − 1 {\displaystyle \omega _{n-1}}

1470-400: The classical asymptotic methods, one "deforms" Riemann–Hilbert problems which are not explicitly solvable to problems that are. The so-called "nonlinear" method of stationary phase is due to Deift & Zhou (1993) , expanding on a previous idea by Its (1982) and Manakov (1974) and using technical background results from Beals & Coifman (1984) and Zhou (1989) . A crucial ingredient of

1519-454: The complex plane), a Riemann–Hilbert factorization problem is the following. Given a matrix function G ( t ) {\displaystyle G(t)} defined on the contour Σ {\displaystyle \Sigma } , find a holomorphic matrix function M ( z ) {\displaystyle M(z)} defined on the complement of Σ {\displaystyle \Sigma } , such that

Riemann–Hilbert problem - Misplaced Pages Continue

1568-513: The conformal map of a harmonic function is also harmonic, the Poisson kernel carries over to the upper half-plane. In this case, the Poisson integral equation takes the form u ( x + i y ) = ∫ − ∞ ∞ P y ( x − t ) f ( t ) d t , y > 0. {\displaystyle u(x+iy)=\int _{-\infty }^{\infty }P_{y}(x-t)f(t)\,dt,\qquad y>0.} The kernel itself

1617-515: The equation for t ∈ Σ {\displaystyle t\in \Sigma } , where a ( t ) {\displaystyle a(t)} , b ( t ) {\displaystyle b(t)} and c ( t ) {\displaystyle c(t)} are given real-valued functions. For example, in the special case where a = 1 , b = 0 {\displaystyle a=1,b=0} and Σ {\displaystyle \Sigma }

1666-478: The external field, was done in Kamvissis & Rakhmanov (2005) ; the contour arising is an "S-curve", as defined and studied in the 1980s by Herbert R. Stahl, Andrei A. Gonchar and Evguenii A Rakhmanov. An alternative asymptotic analysis of Riemann–Hilbert factorization problems is provided in McLaughlin & Miller (2006) , especially convenient when jump matrices do not have analytic extensions. Their method

1715-419: The following two conditions are satisfied In the simplest case G ( t ) {\displaystyle G(t)} is smooth and integrable. In more complicated cases it could have singularities. The limits M + {\displaystyle M_{+}} and M − {\displaystyle M_{-}} could be classical and continuous or they could be taken in

1764-467: The infinite periodic Toda lattice under a "short range" perturbation (for example a perturbation of a finite number of particles). Most Riemann–Hilbert factorization problems studied in the literature are 2-dimensional, i.e., the unknown matrices are of dimension 2. Higher-dimensional problems have been studied by Arno Kuijlaars and collaborators, see e.g. Kuijlaars & López (2015) . Smoothness Too Many Requests If you report this error to

1813-775: The problem attempted to find a pair of analytic functions M + ( t ) {\displaystyle M_{+}(t)} and M − ( t ) {\displaystyle M_{-}(t)} on the inside and outside, respectively, of the curve Σ {\displaystyle \Sigma } , such that for t ∈ Σ {\displaystyle t\in \Sigma } one has where α ( t ) {\displaystyle \alpha (t)} , β ( t ) {\displaystyle \beta (t)} and γ ( t ) {\displaystyle \gamma (t)} are given complex-valued functions (no longer just complex conjugates). In

1862-457: The problem is well-posed. Suppose D {\displaystyle D} is some simply connected domain of the complex z {\displaystyle z} plane . Then the scalar equation is a generalization of a Riemann-Hilbert problem, called the DBAR problem (or ∂ ¯ {\displaystyle {\overline {\partial }}} problem ). It

1911-409: The problem reduces to finding a pair of analytic functions M + ( z ) {\displaystyle M_{+}(z)} and M − ( z ) {\displaystyle M_{-}(z)} on the inside and outside, respectively, of the unit disk , so that on the unit circle and, moreover, so that the condition at infinity holds: Hilbert's generalization of

1960-516: The real line, u can be understood as a harmonic extension of f into the upper half-plane. In analogy to the situation for the disk, when u is holomorphic in the upper half-plane, then u is an element of the Hardy space, H p , {\displaystyle H^{p},} and in particular, ‖ u ‖ H p = ‖ f ‖ L p {\displaystyle \|u\|_{H^{p}}=\|f\|_{L^{p}}} Thus, again,

2009-530: The resulting product { r f k } gives the Abel means A r f of f : A r f ( e 2 π i x ) = ∑ k ∈ Z f k r | k | e 2 π i k x . {\displaystyle A_{r}f(e^{2\pi ix})=\sum _{k\in \mathbb {Z} }f_{k}r^{|k|}e^{2\pi ikx}.} Rearranging this absolutely convergent series shows that f

Riemann–Hilbert problem - Misplaced Pages Continue

2058-426: The small dispersion limit of KdV equation has in fact provided the basis for the analysis of most of the work concerning "real" orthogonal polynomials (i.e. with the orthogonality condition defined on the real line) and Hermitian random matrices. Perhaps the most sophisticated extension of the theory so far is the one applied to the "non self-adjoint" case, i.e. when the underlying Lax operator (the first component of

2107-549: The solution of a Riemann–Hilbert factorization problem is unique (an easy application of Liouville's theorem (complex analysis) ), the Sokhotski–Plemelj theorem gives the solution. We get and therefore which has a branch cut at contour Σ {\displaystyle \Sigma } . Check: therefore, CAVEAT 1: If the problem is not scalar one cannot easily take logarithms. In general explicit solutions are very rare. CAVEAT 2: The boundedness (or at least

2156-486: The solution of the DBAR problem is integrated over the entire complex plane; denoted by R 2 {\displaystyle \mathbb {R} ^{2}} , and where the wedge product is defined as If a function M ( z ) {\displaystyle M(z)} is holomorphic in some complex region R {\displaystyle R} , then in R {\displaystyle R} . For generalized analytic functions, this equation

2205-443: The three problems above (say, as time goes to infinity, or as the dispersion coefficient goes to zero, or as the polynomial degree goes to infinity, or as the size of the permutation goes to infinity). There exists a method for extracting the asymptotic behavior of solutions of Riemann–Hilbert problems, analogous to the method of stationary phase and the method of steepest descent applicable to exponential integrals. By analogy with

2254-873: The unit disc is given by P r ( θ ) = ∑ n = − ∞ ∞ r | n | e i n θ = 1 − r 2 1 − 2 r cos ⁡ θ + r 2 = Re ⁡ ( 1 + r e i θ 1 − r e i θ ) ,       0 ≤ r < 1. {\displaystyle P_{r}(\theta )=\sum _{n=-\infty }^{\infty }r^{|n|}e^{in\theta }={\frac {1-r^{2}}{1-2r\cos \theta +r^{2}}}=\operatorname {Re} \left({\frac {1+re^{i\theta }}{1-re^{i\theta }}}\right),\ \ \ 0\leq r<1.} This can be thought of in two ways: either as

2303-477: The unit disk, and the unit circle. The space of functions that are the limits on T of functions in H ( z ) may be called H ( T ). It is a closed subspace of L ( T ) (at least for p  ≥ 1). Since L ( T ) is a Banach space (for 1 ≤  p  ≤ ∞), so is H ( T ). The unit disk may be conformally mapped to the upper half-plane by means of certain Möbius transformations . Since

2352-714: The upper half-space appears naturally as the Fourier transform of the Abel transform in which t assumes the role of an auxiliary parameter. To wit, P ( t , x ) = F ( K ( t , ⋅ ) ) ( x ) = ∫ R n e − 2 π t | ξ | e − 2 π i ξ ⋅ x d ξ . {\displaystyle P(t,x)={\mathcal {F}}(K(t,\cdot ))(x)=\int _{\mathbb {R} ^{n}}e^{-2\pi t|\xi |}e^{-2\pi i\xi \cdot x}\,d\xi .} In particular, it

2401-458: Was inspired by work of Lax, Levermore and Venakides, who reduced the analysis of the small dispersion limit of the KdV equation to the analysis of a maximization problem for a logarithmic potential under some external field: a variational problem of "electrostatic" type (see Lax & Levermore (1983) ). The g-function is the logarithmic transform of the maximizing "equilibrium" measure. The analysis of

#164835