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Bergman Kernel
In the mathematical study of several complex variables, the Bergman kernel, named after Stefan Bergman, is the reproducing kernel for the Hilbert space (RKHS) of all square integrable holomorphic functions on a domain ''D'' in C''n''. In detail, let L2(''D'') be the Hilbert space of square integrable functions on ''D'', and let ''L''2,''h''(''D'') denote the subspace consisting of holomorphic functions in L2(''D''): that is, :L^(D) = L^2(D)\cap H(D) where ''H''(''D'') is the space of holomorphic functions in ''D''. Then ''L''2,''h''(''D'') is a Hilbert space: it is a closed linear subspace of ''L''2(''D''), and therefore complete in its own right. This follows from the fundamental estimate, that for a holomorphic square-integrable function ''ƒ'' in ''D'' for every compact subset ''K'' of ''D''. Thus convergence of a sequence of holomorphic functions in ''L''2(''D'') implies also compact convergence, and so the limit function is also holomorphic. Another consequence ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complete Metric Space
In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in . Intuitively, a space is complete if there are no "points missing" from it (inside or at the boundary). For instance, the set of rational numbers is not complete, because e.g. \sqrt is "missing" from it, even though one can construct a Cauchy sequence of rational numbers that converges to it (see further examples below). It is always possible to "fill all the holes", leading to the ''completion'' of a given space, as explained below. Definition Cauchy sequence A sequence x_1, x_2, x_3, \ldots in a metric space (X, d) is called Cauchy if for every positive real number r > 0 there is a positive integer N such that for all positive integers m, n > N, d\left(x_m, x_n\right) < r. Complete space A metric space is complete if any of the following equivalent conditions are satisfied: :#Every |
Szegő Kernel
In the mathematical study of several complex variables, the Szegő kernel is an integral kernel that gives rise to a reproducing kernel on a natural Hilbert space of holomorphic functions. It is named for its discoverer, the Hungarian mathematician Gábor Szegő. Let Ω be a bounded domain in C''n'' with ''C''2 boundary, and let ''A''(Ω) denote the space of all holomorphic functions in Ω that are continuous on \overline. Define the Hardy space ''H''2(∂Ω) to be the closure in ''L''2(∂Ω) of the restrictions of elements of ''A''(Ω) to the boundary. The Poisson integral implies that each element ''ƒ'' of ''H''2(∂Ω) extends to a holomorphic function ''Pƒ'' in Ω. Furthermore, for each ''z'' ∈ Ω, the map :f\mapsto Pf(z) defines a continuous linear functional on ''H''2(∂Ω). By the Riesz representation theorem, this linear functional is represented by a kernel ''k''''z'', which is to say :Pf(z ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bergman Space
In complex analysis, functional analysis and operator theory, a Bergman space, named after Stefan Bergman, is a function space of holomorphic functions in a domain ''D'' of the complex plane that are sufficiently well-behaved at the boundary that they are absolutely integrable. Specifically, for , the Bergman space is the space of all holomorphic functions f in ''D'' for which the p-norm is finite: :\, f\, _ := \left(\int_D , f(x+iy), ^p\,\mathrm dx\,\mathrm dy\right)^ < \infty. The quantity is called the ''norm'' of the function ; it is a true if . Thus is the subspace of holomorphic functions that are in the space L''p''(''D''). The Bergman spaces are |
Bergman Metric
In differential geometry, the Bergman metric is a Hermitian metric that can be defined on certain types of complex manifold. It is so called because it is derived from the Bergman kernel, both of which are named after Stefan Bergman. Definition Let G \subset ^n be a domain and let K(z,w) be the Bergman kernel on ''G''. We define a Hermitian metric on the tangent bundle T_z^n by : g_ (z) := \frac \log K(z,z) , for z \in G. Then the length of a tangent vector \xi \in T_z^n is given by :\left\vert \xi \right\vert_:=\sqrt. This metric is called the Bergman metric on ''G''. The length of a (piecewise) ''C''1 curve \gamma \colon ,1\to ^n is then computed as : \ell (\gamma) = \int_0^1 \left\vert \frac(t) \right\vert_ dt . The distance d_G(p,q) of two points p,q \in G is then defined as : d_G(p,q):= \inf \ . The distance ''dG'' is called the ''Bergman distance''. The Bergman metric is in fact a positive definite matrix at each point if ''G'' is a bounded domain. More importantly ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riesz Representation Theorem
:''This article describes a theorem concerning the dual of a Hilbert space. For the theorems relating linear functionals to measures, see Riesz–Markov–Kakutani representation theorem.'' The Riesz representation theorem, sometimes called the Riesz–Fréchet representation theorem after Frigyes Riesz and Maurice René Fréchet, establishes an important connection between a Hilbert space and its continuous dual space. If the underlying field is the real numbers, the two are isometrically isomorphic; if the underlying field is the complex numbers, the two are isometrically anti-isomorphic. The (anti-) isomorphism is a particular natural isomorphism. Preliminaries and notation Let H be a Hilbert space over a field \mathbb, where \mathbb is either the real numbers \R or the complex numbers \Complex. If \mathbb = \Complex (resp. if \mathbb = \R) then H is called a (resp. a ). Every real Hilbert space can be extended to be a dense subset of a unique (up to bijective isometry) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Linear Functional
In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces. An operator between two normed spaces is a bounded linear operator if and only if it is a continuous linear operator. Continuous linear operators Characterizations of continuity Suppose that F : X \to Y is a linear operator between two topological vector spaces (TVSs). The following are equivalent: F is continuous. F is continuous at some point x \in X. F is continuous at the origin in X. if Y is locally convex then this list may be extended to include: for every continuous seminorm q on Y, there exists a continuous seminorm p on X such that q \circ F \leq p. if X and Y are both Hausdorff locally convex spaces then this list may be extended to include: F is weakly continuous and its transpose ^t F : Y^ \to X^ maps equicontinuous subsets of Y^ to equicontinuous subsets of X^ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compact Convergence
In mathematics compact convergence (or uniform convergence on compact sets) is a type of convergence that generalizes the idea of uniform convergence. It is associated with the compact-open topology. Definition Let (X, \mathcal) be a topological space and (Y,d_) be a metric space. A sequence of functions :f_ : X \to Y, n \in \mathbb, is said to converge compactly as n \to \infty to some function f : X \to Y if, for every compact set K \subseteq X, :f_, _ \to f, _ uniformly on K as n \to \infty. This means that for all compact K \subseteq X, :\lim_ \sup_ d_ \left( f_ (x), f(x) \right) = 0. Examples * If X = (0, 1) \subseteq \mathbb and Y = \mathbb with their usual topologies, with f_ (x) := x^, then f_ converges compactly to the constant function with value 0, but not uniformly. * If X=(0,1], Y=\R and f_n(x)=x^n, then f_n converges pointwise convergence, pointwise to the function that is zero on (0,1) and one at 1, but the sequence does not converge compactly. * A very po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compact Set
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i.e. that the space not exclude any ''limiting values'' of points. For example, the open interval (0,1) would not be compact because it excludes the limiting values of 0 and 1, whereas the closed interval ,1would be compact. Similarly, the space of rational numbers \mathbb is not compact, because it has infinitely many "punctures" corresponding to the irrational numbers, and the space of real numbers \mathbb is not compact either, because it excludes the two limiting values +\infty and -\infty. However, the ''extended'' real number line ''would'' be compact, since it contains both infinities. There are many ways to make this heuristic notion precise. These ways usually agree in a metric space, but may not be equivalent in other topologic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closed Set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a closed set is a set which is closed under the limit operation. This should not be confused with a closed manifold. Equivalent definitions By definition, a subset A of a topological space (X, \tau) is called if its complement X \setminus A is an open subset of (X, \tau); that is, if X \setminus A \in \tau. A set is closed in X if and only if it is equal to its closure in X. Equivalently, a set is closed if and only if it contains all of its limit points. Yet another equivalent definition is that a set is closed if and only if it contains all of its boundary points. Every subset A \subseteq X is always contained in its (topological) closure in X, which is denoted by \operatorname_X A; that is, if A \subseteq X then A \subseteq \oper ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Several Complex Variables
The theory of functions of several complex variables is the branch of mathematics dealing with complex-valued functions. The name of the field dealing with the properties of function of several complex variables is called several complex variables (and analytic space), that has become a common name for that whole field of study and Mathematics Subject Classification has, as a top-level heading. A function f:(z_1,z_2, \ldots, z_n) \rightarrow f(z_1,z_2, \ldots, z_n) is -tuples of complex numbers, classically studied on the complex coordinate space \Complex^n. As in complex analysis of functions of one variable, which is the case , the functions studied are ''holomorphic'' or ''complex analytic'' so that, locally, they are power series in the variables . Equivalently, they are locally uniform limits of polynomials; or locally square-integrable solutions to the -dimensional Cauchy–Riemann equations. For one complex variable, every domainThat is an open connected subset. (D \subs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lp Space
In mathematics, the spaces are function spaces defined using a natural generalization of the Norm (mathematics)#p-norm, -norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue , although according to the Nicolas Bourbaki, Bourbaki group they were first introduced by Frigyes Riesz . spaces form an important class of Banach spaces in functional analysis, and of topological vector spaces. Because of their key role in the mathematical analysis of measure and probability spaces, Lebesgue spaces are used also in the theoretical discussion of problems in physics, statistics, economics, finance, engineering, and other disciplines. Applications Statistics In statistics, measures of central tendency and statistical dispersion, such as the mean, median, and standard deviation, are defined in terms of metrics, and measures of central tendency can be characterized as Central tendency#Solutions to variational problems, solutions to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
