Misplaced Pages

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100-433: 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

200-453: A k th derivative that is continuous in its domain. A function of class C ∞ {\displaystyle C^{\infty }} or C ∞ {\displaystyle C^{\infty }} -function (pronounced C-infinity function ) is an infinitely differentiable function , that is, a function that has derivatives of all orders (this implies that all these derivatives are continuous). Generally,

300-419: 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

400-702: A curve: In general, G n {\displaystyle G^{n}} continuity exists if the curves can be reparameterized to have C n {\displaystyle C^{n}} (parametric) continuity. A reparametrization of the curve is geometrically identical to the original; only the parameter is affected. Equivalently, two vector functions f ( t ) {\displaystyle f(t)} and g ( t ) {\displaystyle g(t)} such that f ( 1 ) = g ( 0 ) {\displaystyle f(1)=g(0)} have G n {\displaystyle G^{n}} continuity at

500-405: A film, higher orders of parametric continuity are required. The various order of parametric continuity can be described as follows: A curve or surface can be described as having G n {\displaystyle G^{n}} continuity, with n {\displaystyle n} being the increasing measure of smoothness. Consider the segments either side of a point on

600-806: 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}

700-444: A function. Consider an open set U {\displaystyle U} on the real line and a function f {\displaystyle f} defined on U {\displaystyle U} with real values. Let k be a non-negative integer . The function f {\displaystyle f} is said to be of differentiability class C k {\displaystyle C^{k}} if

800-719: A map f : M → R {\displaystyle f:M\to \mathbb {R} } is smooth on M {\displaystyle M} if for all p ∈ M {\displaystyle p\in M} there exists a chart ( U , ϕ ) ∈ U , {\displaystyle (U,\phi )\in {\mathfrak {U}},} such that p ∈ U , {\displaystyle p\in U,} and f ∘ ϕ − 1 : ϕ ( U ) → R {\displaystyle f\circ \phi ^{-1}:\phi (U)\to \mathbb {R} }

900-461: A sphere at its corners and quarter-cylinders along its edges. If an editable curve with G 2 {\displaystyle G^{2}} continuity is required, then cubic splines are typically chosen; these curves are frequently used in industrial design . While all analytic functions are "smooth" (i.e. have all derivatives continuous) on the set on which they are analytic, examples such as bump functions (mentioned above) show that

1000-428: Is D {\displaystyle D} , and m = 0 , 1 , … , k {\displaystyle m=0,1,\dots ,k} . The set of C ∞ {\displaystyle C^{\infty }} functions over D {\displaystyle D} also forms a Fréchet space. One uses the same seminorms as above, except that m {\displaystyle m}

1100-471: 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

SECTION 10

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1200-480: Is a Fréchet vector space , with the countable family of seminorms p K , m = sup x ∈ K | f ( m ) ( x ) | {\displaystyle p_{K,m}=\sup _{x\in K}\left|f^{(m)}(x)\right|} where K {\displaystyle K} varies over an increasing sequence of compact sets whose union

1300-865: 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

1400-674: Is a smooth function from R n . {\displaystyle \mathbb {R} ^{n}.} Smooth maps between manifolds induce linear maps between tangent spaces : for F : M → N {\displaystyle F:M\to N} , at each point the pushforward (or differential) maps tangent vectors at p {\displaystyle p} to tangent vectors at F ( p ) {\displaystyle F(p)} : F ∗ , p : T p M → T F ( p ) N , {\displaystyle F_{*,p}:T_{p}M\to T_{F(p)}N,} and on

1500-466: Is a smooth function from a neighborhood of ϕ ( p ) {\displaystyle \phi (p)} in R m {\displaystyle \mathbb {R} ^{m}} to R {\displaystyle \mathbb {R} } (all partial derivatives up to a given order are continuous). Smoothness can be checked with respect to any chart of the atlas that contains p , {\displaystyle p,} since

1600-504: Is allowed to range over all non-negative integer values. The above spaces occur naturally in applications where functions having derivatives of certain orders are necessary; however, particularly in the study of partial differential equations , it can sometimes be more fruitful to work instead with the Sobolev spaces . The terms parametric continuity ( C ) and geometric continuity ( G ) were introduced by Brian Barsky , to show that

1700-413: Is at least in the class C k − 1 {\displaystyle C^{k-1}} since f ′ , f ″ , … , f ( k − 1 ) {\displaystyle f',f'',\dots ,f^{(k-1)}} are continuous on U . {\displaystyle U.} The function f {\displaystyle f}

1800-612: 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

1900-400: Is both infinitely differentiable and analytic on that set . Smooth functions with given closed support are used in the construction of smooth partitions of unity (see partition of unity and topology glossary ); these are essential in the study of smooth manifolds , for example to show that Riemannian metrics can be defined globally starting from their local existence. A simple case

2000-575: 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

2100-480: 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

SECTION 20

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2200-447: Is constrained to be positive. In the case n = 1 {\displaystyle n=1} , this reduces to f ′ ( 1 ) ≠ 0 {\displaystyle f'(1)\neq 0} and f ′ ( 1 ) = k g ′ ( 0 ) {\displaystyle f'(1)=kg'(0)} , for a scalar k > 0 {\displaystyle k>0} (i.e.,

2300-501: Is contained in C k − 1 {\displaystyle C^{k-1}} for every k > 0 , {\displaystyle k>0,} and there are examples to show that this containment is strict ( C k ⊊ C k − 1 {\displaystyle C^{k}\subsetneq C^{k-1}} ). The class C ∞ {\displaystyle C^{\infty }} of infinitely differentiable functions,

2400-468: Is continuous, but not differentiable at x = 0 , so it is of class C , but not of class C . For each even integer k , the function f ( x ) = | x | k + 1 {\displaystyle f(x)=|x|^{k+1}} is continuous and k times differentiable at all x . At x = 0 , however, f {\displaystyle f} is not ( k + 1) times differentiable, so f {\displaystyle f}

2500-480: Is continuous; such functions are called continuously differentiable . Thus, a C 1 {\displaystyle C^{1}} function is exactly a function whose derivative exists and is of class C 0 . {\displaystyle C^{0}.} In general, the classes C k {\displaystyle C^{k}} can be defined recursively by declaring C 0 {\displaystyle C^{0}} to be

2600-1075: Is differentiable but its derivative is unbounded on a compact set . Therefore, h {\displaystyle h} is an example of a function that is differentiable but not locally Lipschitz continuous . The exponential function e x {\displaystyle e^{x}} is analytic, and hence falls into the class C . The trigonometric functions are also analytic wherever they are defined, because they are linear combinations of complex exponential functions e i x {\displaystyle e^{ix}} and e − i x {\displaystyle e^{-ix}} . The bump function f ( x ) = { e − 1 1 − x 2  if  | x | < 1 , 0  otherwise  {\displaystyle f(x)={\begin{cases}e^{-{\frac {1}{1-x^{2}}}}&{\text{ if }}|x|<1,\\0&{\text{ otherwise }}\end{cases}}}

2700-756: Is differentiable, with derivative g ′ ( x ) = { − cos ⁡ ( 1 x ) + 2 x sin ⁡ ( 1 x ) if  x ≠ 0 , 0 if  x = 0. {\displaystyle g'(x)={\begin{cases}-{\mathord {\cos \left({\tfrac {1}{x}}\right)}}+2x\sin \left({\tfrac {1}{x}}\right)&{\text{if }}x\neq 0,\\0&{\text{if }}x=0.\end{cases}}} Because cos ⁡ ( 1 / x ) {\displaystyle \cos(1/x)} oscillates as x → 0, g ′ ( x ) {\displaystyle g'(x)}

2800-474: 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

2900-732: 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

3000-456: 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

3100-551: Is not continuous at zero. Therefore, g ( x ) {\displaystyle g(x)} is differentiable but not of class C . The function h ( x ) = { x 4 / 3 sin ⁡ ( 1 x ) if  x ≠ 0 , 0 if  x = 0 {\displaystyle h(x)={\begin{cases}x^{4/3}\sin {\left({\tfrac {1}{x}}\right)}&{\text{if }}x\neq 0,\\0&{\text{if }}x=0\end{cases}}}

Riemann–Hilbert problem - Misplaced Pages Continue

3200-396: Is of class C k {\displaystyle C^{k}} on U {\displaystyle U} if the k {\displaystyle k} -th order Fréchet derivative of f {\displaystyle f} exists and is continuous at every point of U {\displaystyle U} . The function f {\displaystyle f}

3300-453: Is of class C , but not of class C where j > k . The function g ( x ) = { x 2 sin ⁡ ( 1 x ) if  x ≠ 0 , 0 if  x = 0 {\displaystyle g(x)={\begin{cases}x^{2}\sin {\left({\tfrac {1}{x}}\right)}&{\text{if }}x\neq 0,\\0&{\text{if }}x=0\end{cases}}}

3400-1170: 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

3500-1570: Is said to be of class C k {\displaystyle C^{k}} on U {\displaystyle U} , for a positive integer k {\displaystyle k} , if all partial derivatives ∂ α f ∂ x 1 α 1 ∂ x 2 α 2 ⋯ ∂ x n α n ( y 1 , y 2 , … , y n ) {\displaystyle {\frac {\partial ^{\alpha }f}{\partial x_{1}^{\alpha _{1}}\,\partial x_{2}^{\alpha _{2}}\,\cdots \,\partial x_{n}^{\alpha _{n}}}}(y_{1},y_{2},\ldots ,y_{n})} exist and are continuous, for every α 1 , α 2 , … , α n {\displaystyle \alpha _{1},\alpha _{2},\ldots ,\alpha _{n}} non-negative integers, such that α = α 1 + α 2 + ⋯ + α n ≤ k {\displaystyle \alpha =\alpha _{1}+\alpha _{2}+\cdots +\alpha _{n}\leq k} , and every ( y 1 , y 2 , … , y n ) ∈ U {\displaystyle (y_{1},y_{2},\ldots ,y_{n})\in U} . Equivalently, f {\displaystyle f}

3600-401: Is said to be infinitely differentiable , smooth , or of class C ∞ , {\displaystyle C^{\infty },} if it has derivatives of all orders on U . {\displaystyle U.} (So all these derivatives are continuous functions over U . {\displaystyle U.} ) The function f {\displaystyle f}

3700-767: Is said to be smooth if for all x ∈ X {\displaystyle x\in X} there is an open set U ⊆ M {\displaystyle U\subseteq M} with x ∈ U {\displaystyle x\in U} and a smooth function F : U → N {\displaystyle F:U\to N} such that F ( p ) = f ( p ) {\displaystyle F(p)=f(p)} for all p ∈ U ∩ X . {\displaystyle p\in U\cap X.} Poisson kernel In mathematics, and specifically in potential theory ,

3800-408: Is said to be of class C ω , {\displaystyle C^{\omega },} or analytic , if f {\displaystyle f} is smooth (i.e., f {\displaystyle f} is in the class C ∞ {\displaystyle C^{\infty }} ) and its Taylor series expansion around any point in its domain converges to

3900-1062: Is said to be of class C k {\displaystyle C^{k}} on U {\displaystyle U} , for a positive integer k {\displaystyle k} , if all of its components f i ( x 1 , x 2 , … , x n ) = ( π i ∘ f ) ( x 1 , x 2 , … , x n ) = π i ( f ( x 1 , x 2 , … , x n ) )  for  i = 1 , 2 , 3 , … , m {\displaystyle f_{i}(x_{1},x_{2},\ldots ,x_{n})=(\pi _{i}\circ f)(x_{1},x_{2},\ldots ,x_{n})=\pi _{i}(f(x_{1},x_{2},\ldots ,x_{n})){\text{ for }}i=1,2,3,\ldots ,m} are of class C k {\displaystyle C^{k}} , where π i {\displaystyle \pi _{i}} are

4000-674: Is said to be of class C {\displaystyle C} or C 0 {\displaystyle C^{0}} if it is continuous on U {\displaystyle U} . Functions of class C 1 {\displaystyle C^{1}} are also said to be continuously differentiable . A function f : U ⊂ R n → R m {\displaystyle f:U\subset \mathbb {R} ^{n}\to \mathbb {R} ^{m}} , defined on an open set U {\displaystyle U} of R n {\displaystyle \mathbb {R} ^{n}} ,

4100-556: Is said to be of class C {\displaystyle C} or C 0 {\displaystyle C^{0}} if it is continuous, or equivalently, if all components f i {\displaystyle f_{i}} are continuous, on U {\displaystyle U} . Let D {\displaystyle D} be an open subset of the real line. The set of all C k {\displaystyle C^{k}} real-valued functions defined on D {\displaystyle D}

Riemann–Hilbert problem - Misplaced Pages Continue

4200-449: Is said to be of class C , if d k s d t k {\displaystyle \textstyle {\frac {d^{k}s}{dt^{k}}}} exists and is continuous on [ 0 , 1 ] {\displaystyle [0,1]} , where derivatives at the end-points 0 {\displaystyle 0} and 1 {\displaystyle 1} are taken to be one sided derivatives (from

4300-778: Is smooth if, for every p ∈ M , {\displaystyle p\in M,} there is a chart ( U , ϕ ) {\displaystyle (U,\phi )} containing p , {\displaystyle p,} and a chart ( V , ψ ) {\displaystyle (V,\psi )} containing F ( p ) {\displaystyle F(p)} such that F ( U ) ⊂ V , {\displaystyle F(U)\subset V,} and ψ ∘ F ∘ ϕ − 1 : ϕ ( U ) → ψ ( V ) {\displaystyle \psi \circ F\circ \phi ^{-1}:\phi (U)\to \psi (V)}

4400-483: Is smooth, so of class C , but it is not analytic at x = ±1 , and hence is not of class C . The function f is an example of a smooth function with compact support . A function f : U ⊂ R n → R {\displaystyle f:U\subset \mathbb {R} ^{n}\to \mathbb {R} } defined on an open set U {\displaystyle U} of R n {\displaystyle \mathbb {R} ^{n}}

4500-670: Is that of a bump function on the real line, that is, a smooth function f that takes the value 0 outside an interval [ a , b ] and such that f ( x ) > 0  for  a < x < b . {\displaystyle f(x)>0\quad {\text{ for }}\quad a<x<b.\,} Given a number of overlapping intervals on the line, bump functions can be constructed on each of them, and on semi-infinite intervals ( − ∞ , c ] {\displaystyle (-\infty ,c]} and [ d , + ∞ ) {\displaystyle [d,+\infty )} to cover

4600-510: 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 )

4700-466: 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

4800-557: 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

4900-476: Is the intersection of the classes C k {\displaystyle C^{k}} as k {\displaystyle k} varies over the non-negative integers. The function f ( x ) = { x if  x ≥ 0 , 0 if  x < 0 {\displaystyle f(x)={\begin{cases}x&{\mbox{if }}x\geq 0,\\0&{\text{if }}x<0\end{cases}}}

5000-471: 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

5100-913: 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

SECTION 50

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5200-890: 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

5300-616: 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}

5400-637: 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

5500-447: 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

5600-529: The Poisson kernel is an integral kernel , used for solving the two-dimensional Laplace equation , given Dirichlet boundary conditions on 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,

5700-468: 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

5800-400: The smoothness of a function is a property measured by the number of continuous derivatives ( differentiability class) it has over its domain . A function of class C k {\displaystyle C^{k}} is a function of smoothness at least k ; that is, a function of class C k {\displaystyle C^{k}} is a function that has

5900-632: 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} } ,

6000-488: 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

6100-461: 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

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6200-481: 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

6300-945: 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}}

6400-402: The body has G 2 {\displaystyle G^{2}} continuity. A rounded rectangle (with ninety degree circular arcs at the four corners) has G 1 {\displaystyle G^{1}} continuity, but does not have G 2 {\displaystyle G^{2}} continuity. The same is true for a rounded cube , with octants of

6500-400: 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

6600-454: 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

6700-513: 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

6800-478: The converse is not true for functions on the reals: there exist smooth real functions that are not analytic. Simple examples of functions that are smooth but not analytic at any point can be made by means of Fourier series ; another example is the Fabius function . Although it might seem that such functions are the exception rather than the rule, it turns out that the analytic functions are scattered very thinly among

6900-1004: The definition of Poisson kernels are often extended to n -dimensional problems. In the complex plane , the Poisson kernel for 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

7000-432: The derivatives f ′ , f ″ , … , f ( k ) {\displaystyle f',f'',\dots ,f^{(k)}} exist and are continuous on U . {\displaystyle U.} If f {\displaystyle f} is k {\displaystyle k} -differentiable on U , {\displaystyle U,} then it

7100-599: The differential does not vanish on the preimage) are manifolds; this is the preimage theorem . Similarly, pushforwards along embeddings are manifolds. There is a corresponding notion of smooth map for arbitrary subsets of manifolds. If f : X → Y {\displaystyle f:X\to Y} is a function whose domain and range are subsets of manifolds X ⊆ M {\displaystyle X\subseteq M} and Y ⊆ N {\displaystyle Y\subseteq N} respectively. f {\displaystyle f}

7200-436: The direction, but not necessarily the magnitude, of the two vectors is equal). While it may be obvious that a curve would require G 1 {\displaystyle G^{1}} continuity to appear smooth, for good aesthetics , such as those aspired to in architecture and sports car design, higher levels of geometric continuity are required. For example, reflections in a car body will not appear smooth unless

7300-515: 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 }

7400-415: The examples we come up with at first thought (algebraic/rational numbers and analytic functions) are far better behaved than the majority of cases: the transcendental numbers and nowhere analytic functions have full measure (their complements are meagre). The situation thus described is in marked contrast to complex differentiable functions. If a complex function is differentiable just once on an open set, it

7500-478: 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

7600-419: 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

7700-628: The function in some neighborhood of the point. There exist functions that are smooth but not analytic; C ω {\displaystyle C^{\omega }} is thus strictly contained in C ∞ . {\displaystyle C^{\infty }.} Bump functions are examples of functions with this property. To put it differently, the class C 0 {\displaystyle C^{0}} consists of all continuous functions. The class C 1 {\displaystyle C^{1}} consists of all differentiable functions whose derivative

7800-448: 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 In mathematical analysis ,

7900-1119: The level of the tangent bundle , the pushforward is a vector bundle homomorphism : F ∗ : T M → T N . {\displaystyle F_{*}:TM\to TN.} The dual to the pushforward is the pullback , which "pulls" covectors on N {\displaystyle N} back to covectors on M , {\displaystyle M,} and k {\displaystyle k} -forms to k {\displaystyle k} -forms: F ∗ : Ω k ( N ) → Ω k ( M ) . {\displaystyle F^{*}:\Omega ^{k}(N)\to \Omega ^{k}(M).} In this way smooth functions between manifolds can transport local data , like vector fields and differential forms , from one manifold to another, or down to Euclidean space where computations like integration are well understood. Preimages and pushforwards along smooth functions are, in general, not manifolds without additional assumptions. Preimages of regular points (that is, if

8000-418: The natural projections π i : R m → R {\displaystyle \pi _{i}:\mathbb {R} ^{m}\to \mathbb {R} } defined by π i ( x 1 , x 2 , … , x m ) = x i {\displaystyle \pi _{i}(x_{1},x_{2},\ldots ,x_{m})=x_{i}} . It

8100-592: The point where they meet if they satisfy equations known as Beta-constraints. For example, the Beta-constraints for G 4 {\displaystyle G^{4}} continuity are: where β 2 {\displaystyle \beta _{2}} , β 3 {\displaystyle \beta _{3}} , and β 4 {\displaystyle \beta _{4}} are arbitrary, but β 1 {\displaystyle \beta _{1}}

8200-775: 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

8300-457: 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

8400-409: 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

8500-516: 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,

8600-530: 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

8700-420: The right at 0 {\displaystyle 0} and from the left at 1 {\displaystyle 1} ). As a practical application of this concept, a curve describing the motion of an object with a parameter of time must have C continuity and its first derivative is differentiable—for the object to have finite acceleration. For smoother motion, such as that of a camera's path while making

8800-412: The set of all continuous functions, and declaring C k {\displaystyle C^{k}} for any positive integer k {\displaystyle k} to be the set of all differentiable functions whose derivative is in C k − 1 . {\displaystyle C^{k-1}.} In particular, C k {\displaystyle C^{k}}

8900-426: 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

9000-400: The smooth ones; more rigorously, the analytic functions form a meagre subset of the smooth functions. Furthermore, for every open subset A of the real line, there exist smooth functions that are analytic on A and nowhere else . It is useful to compare the situation to that of the ubiquity of transcendental numbers on the real line. Both on the real line and the set of smooth functions,

9100-451: The smoothness of a curve could be measured by removing restrictions on the speed , with which the parameter traces out the curve. Parametric continuity ( C ) is a concept applied to parametric curves , which describes the smoothness of the parameter's value with distance along the curve. A (parametric) curve s : [ 0 , 1 ] → R n {\displaystyle s:[0,1]\to \mathbb {R} ^{n}}

9200-603: The smoothness requirements on the transition functions between charts ensure that if f {\displaystyle f} is smooth near p {\displaystyle p} in one chart it will be smooth near p {\displaystyle p} in any other chart. If F : M → N {\displaystyle F:M\to N} is a map from M {\displaystyle M} to an n {\displaystyle n} -dimensional manifold N {\displaystyle N} , then F {\displaystyle F}

9300-549: 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

9400-486: 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

9500-416: The term smooth function refers to a C ∞ {\displaystyle C^{\infty }} -function. However, it may also mean "sufficiently differentiable" for the problem under consideration. Differentiability class is a classification of functions according to the properties of their derivatives . It is a measure of the highest order of derivative that exists and is continuous for

9600-443: 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

9700-477: 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

9800-714: 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

9900-748: The whole line, such that the sum of the functions is always 1. From what has just been said, partitions of unity do not apply to holomorphic functions ; their different behavior relative to existence and analytic continuation is one of the roots of sheaf theory. In contrast, sheaves of smooth functions tend not to carry much topological information. Given a smooth manifold M {\displaystyle M} , of dimension m , {\displaystyle m,} and an atlas U = { ( U α , ϕ α ) } α , {\displaystyle {\mathfrak {U}}=\{(U_{\alpha },\phi _{\alpha })\}_{\alpha },} then

10000-458: 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

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