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Bochner's Tube Theorem
In mathematics, Bochner's tube theorem (named for Salomon Bochner) shows that every function holomorphic on a tube domain in \mathbb^n can be extended to the convex hull of this domain. Theorem Let \omega \subset \mathbb^n be a connected open set. Then every function f(z) holomorphic on the tube domain \Omega = \omega+i \mathbb^n can be extended to a function holomorphic on the convex hull \operatorname(\Omega). A classic reference is (Theorem 9). See also and for other proofs. Generalizations The generalized version of this theorem was first proved by Kazlow (1979), also proved by Boivin and Dwilewicz (1998). under more less complicated hypothese. Theorem Let \omega be a connected submanifold of \mathbb^n of class-C^2. Then every continuous CR function CR or Cr may refer to: In business * Conversion rate, in marketing * Credit Record, in accounting * Crown Royal, a brand of Canadian whisky Organizations Religious organizations * Celtic Reconstructionism, a form o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Salomon Bochner
Salomon Bochner (20 August 1899 – 2 May 1982) was a Galizien-born mathematician, known for work in mathematical analysis, probability theory and differential geometry. Life He was born into a Jewish family in Podgórze (near Kraków), then Austria-Hungary, now Poland. Fearful of a Russian invasion in Galicia at the beginning of World War I in 1914, his family moved to Germany, seeking greater security. Bochner was educated at a Berlin gymnasium (secondary school), and then at the University of Berlin. There, he was a student of Erhard Schmidt, writing a dissertation involving what would later be called the Bergman kernel. Shortly after this, he left the academy to help his family during the escalating inflation. After returning to mathematical research, he lectured at the University of Munich from 1924 to 1933. His academic career in Germany ended after the Nazis came to power in 1933, and he left for a position at Princeton University. He was a visiting scholar at th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tube Domain
In mathematics, a tube domain is a generalization of the notion of a vertical strip (or half-plane) in the complex plane to several complex variables. A strip can be thought of as the collection of complex numbers whose real part lie in a given subset of the real line and whose imaginary part is unconstrained; likewise, a tube is the set of complex vectors whose real part is in some given collection of real vectors, and whose imaginary part is unconstrained. Tube domains are domains of the Laplace transform of a function of several real variables (see multidimensional Laplace transform). Hardy spaces on tubes can be defined in a manner in which a version of the Paley–Wiener theorem from one variable continues to hold, and characterizes the elements of Hardy spaces as the Laplace transforms of functions with appropriate integrability properties. Tubes over convex sets are domains of holomorphy. The Hardy spaces on tubes over convex cones have an especially rich structure, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Hull
In geometry, the convex hull, convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, or equivalently as the set of all convex combinations of points in the subset. For a Bounded set, bounded subset of the plane, the convex hull may be visualized as the shape enclosed by a rubber band stretched around the subset. Convex hulls of open sets are open, and convex hulls of compact sets are compact. Every compact convex set is the convex hull of its extreme points. The convex hull operator is an example of a closure operator, and every antimatroid can be represented by applying this closure operator to finite sets of points. The algorithmic problems of finding the convex hull of a finite set of points in the plane or other low-dimensional Euclidean spaces, and its projective duality, dual problem of intersecting Half-space (geome ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smoothness
In mathematical analysis, 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 is a function of smoothness at least ; that is, a function of class C^k is a function that has a th derivative that is continuous in its domain. A function of class C^\infty or 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, the term smooth function refers to a C^-function. However, it may also mean "sufficiently differentiable" for the problem under consideration. Differentiability classes 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 a function. Consider an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
CR Function
CR or Cr may refer to: In business * Conversion rate, in marketing * Credit Record, in accounting * Crown Royal, a brand of Canadian whisky Organizations Religious organizations * Celtic Reconstructionism, a form of Polytheism * Congregation of Clerics Regular of the Divine Providence (Theatines), a Roman Catholic religious order * Community of the Resurrection, an Anglican religious order * Congregation of the Resurrection, a Catholic religious order Other organizations * Choose Responsibility, a US non-profit addressing alcohol consumption by young adults * Consumer Reports, an American nonprofit consumer organization dedicated to independent product testing, investigative journalism, consumer-oriented research, public education, and consumer advocacy; publishes a magazine of the same name * College of the Redwoods, a public two-year community college in Humboldt County, California, US * College Republicans, a college branch of the US political party * Czech Radio, a public ... [...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 functions defined on the complex coordinate space \mathbb C^n, that is, -tuples of complex numbers. The name of the field dealing with the properties of these functions is called several complex variables (and analytic space), which the Mathematics Subject Classification has as a top-level heading. 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 \subset \mathbb C), is the domain of holomorphy of some function, in other words every domain has a function for which it is the domain of holomorphy. For several complex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
