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Schur Test
In mathematical analysis, the Schur test, named after German mathematician Issai Schur, is a bound on the L^2\to L^2 operator norm of an integral operator in terms of its Schwartz kernel (see Schwartz kernel theorem). Here is one version. Let X,\,Y be two measurable spaces (such as \mathbb^n). Let \,T be an integral operator with the non-negative Schwartz kernel \,K(x,y), x\in X, y\in Y: :T f(x)=\int_Y K(x,y)f(y)\,dy. If there exist real functions \,p(x)>0 and \,q(y)>0 and numbers \,\alpha,\beta>0 such that : (1)\qquad \int_Y K(x,y)q(y)\,dy\le\alpha p(x) for almost all \,x and : (2)\qquad \int_X p(x)K(x,y)\,dx\le\beta q(y) for almost all \,y, then \,T extends to a continuous operator T:L^2\to L^2 with the operator norm : \Vert T\Vert_ \le\sqrt. Such functions \,p(x), \,q(y) are called the Schur test functions. In the original version, \,T is a matrix and \,\alpha=\beta=1. Common usage and Young's inequality A common usage of the Schur test is to take \,p(x)=q( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (mathematics), series, and analytic functions. These theories are usually studied in the context of Real number, real and Complex number, complex numbers and Function (mathematics), functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry; however, it can be applied to any Space (mathematics), space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space). History Ancient Mathematical analysis formally developed in the 17th century during the Scientific Revolution, but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Issai Schur
Issai Schur (10 January 1875 – 10 January 1941) was a Russian mathematician who worked in Germany for most of his life. He studied at the University of Berlin. He obtained his doctorate in 1901, became lecturer in 1903 and, after a stay at the University of Bonn, professor in 1919. As a student of Ferdinand Georg Frobenius, he worked on group representations (the subject with which he is most closely associated), but also in combinatorics and number theory and even theoretical physics. He is perhaps best known today for his result on the existence of the Schur decomposition and for his work on group representations ( Schur's lemma). Schur published under the name of both I. Schur, and J. Schur, the latter especially in ''Journal für die reine und angewandte Mathematik''. This has led to some confusion. Childhood Issai Schur was born into a Jewish family, the son of the businessman Moses Schur and his wife Golde Schur (née Landau). He was born in Mogilev on the Dnieper Riv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Operator Norm
In mathematics, the operator norm measures the "size" of certain linear operators by assigning each a real number called its . Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces. Introduction and definition Given two normed vector spaces V and W (over the same base field, either the real numbers \R or the complex numbers \Complex), a linear map A : V \to W is continuous if and only if there exists a real number c such that \, Av\, \leq c \, v\, \quad \mbox v\in V. The norm on the left is the one in W and the norm on the right is the one in V. Intuitively, the continuous operator A never increases the length of any vector by more than a factor of c. Thus the image of a bounded set under a continuous operator is also bounded. Because of this property, the continuous linear operators are also known as bounded operators. In order to "measure the size" of A, one can take the infimum of the numbers c such that the abo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral Operator
An integral operator is an operator that involves integration. Special instances are: * The operator of integration itself, denoted by the integral symbol * Integral linear operators, which are linear operators induced by bilinear forms involving integrals * Integral transforms, which are maps between two function space In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a vect ...s, which involve integrals {{mathanalysis-stub Integral calculus ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schwartz Kernel , a surname
{{disambiguation ...
Schwartz may refer to: *Schwartz (surname), a surname (and list of people with the name) *Schwartz (brand), a spice brand * Schwartz's, a delicatessen in Montreal, Quebec, Canada *Schwartz Publishing, an Australian publishing house *"Danny Schwartz", a police detective in the film '' Heat'' portrayed by Jerry Trimble * C. F. Schwartz, Rev, an 18th-century missionary, member of the Church Mission Society, England, sent to India for missionary work *"The Schwartz", a parody of the Force from ''Star Wars'' in the 1987 comedy science-fiction film ''Spaceballs'' See also *Schwarz (other) *Swartz (other) *Schwarcz Schwarcz is a surname. Notable people with the surname include: * Joseph A. Schwarcz, Canadian chemist and writer * June Schwarcz (1918–2015), American artist * Mordechai Schwarcz (1914–1938), Czech-born Jewish police officer in Mandatory Pales ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schwartz Kernel Theorem
In mathematics, the Schwartz kernel theorem is a foundational result in the theory of generalized functions, published by Laurent Schwartz in 1952. It states, in broad terms, that the generalized functions introduced by Schwartz (Schwartz distributions) have a two-variable theory that includes all reasonable bilinear forms on the space \mathcal of test functions. The space \mathcal itself consists of smooth functions of compact support. Statement of the theorem Let X and Y be open sets in \mathbb^n. Every distribution k \in \mathcal'(X \times Y) defines a continuous linear map K \colon \mathcal(Y) \to \mathcal'(X) such that for every u \in \mathcal(X), v \in \mathcal(Y). Conversely, for every such continuous linear map K there exists one and only one distribution k \in \mathcal'(X \times Y) such that () holds. The distribution k is the kernel of the map K. Note Given a distribution k \in \mathcal'(X \times Y) one can always write the linear map K informally as :Kv = \int_ k(\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul Richard Halmos
Paul Richard Halmos ( hu, Halmos Pál; March 3, 1916 – October 2, 2006) was a Hungarian-born American mathematician and statistician who made fundamental advances in the areas of mathematical logic, probability theory, statistics, operator theory, ergodic theory, and functional analysis (in particular, Hilbert spaces). He was also recognized as a great mathematical expositor. He has been described as one of The Martians. Early life and education Born in Hungary into a Jewish family, Halmos arrived in the U.S. at 13 years of age. He obtained his B.A. from the University of Illinois, majoring in mathematics, but fulfilling the requirements for both a math and philosophy degree. He took only three years to obtain the degree, and was only 19 when he graduated. He then began a Ph.D. in philosophy, still at the Champaign–Urbana campus; but, after failing his masters' oral exams, he shifted to mathematics, graduating in 1938. Joseph L. Doob supervised his dissertation, titled ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Measurable Space
In mathematics, a measurable space or Borel space is a basic object in measure theory. It consists of a set and a σ-algebra, which defines the subsets that will be measured. Definition Consider a set X and a σ-algebra \mathcal A on X. Then the tuple (X, \mathcal A) is called a measurable space. Note that in contrast to a measure space A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that i ..., no measure is needed for a measurable space. Example Look at the set: X = \. One possible \sigma-algebra would be: \mathcal A_1 = \. Then \left(X, \mathcal A_1\right) is a measurable space. Another possible \sigma-algebra would be the power set on X: \mathcal A_2 = \mathcal P(X). With this, a second measurable space on the set X is given by \left(X, \mathcal A_2\right). Common measurable ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Almost Everywhere
In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to the concept of measure zero, and is analogous to the notion of ''almost surely'' in probability theory. More specifically, a property holds almost everywhere if it holds for all elements in a set except a subset of measure zero, or equivalently, if the set of elements for which the property holds is conull. In cases where the measure is not complete, it is sufficient that the set be contained within a set of measure zero. When discussing sets of real numbers, the Lebesgue measure is usually assumed unless otherwise stated. The term ''almost everywhere'' is abbreviated ''a.e.''; in older literature ''p.p.'' is used, to stand for the equivalent French language phrase ''presque partout''. A set with full measure is one whose complement ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Operator
In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y. If X and Y are normed vector spaces (a special type of TVS), then L is bounded if and only if there exists some M > 0 such that for all x \in X, \, Lx\, _Y \leq M \, x\, _X. The smallest such M is called the operator norm of L and denoted by \, L\, . A bounded operator between normed spaces is continuous and vice versa. The concept of a bounded linear operator has been extended from normed spaces to all topological vector spaces. Outside of functional analysis, when a function f : X \to Y is called "bounded" then this usually means that its image f(X) is a bounded subset of its codomain. A linear map has this property if and only if it is identically 0. Consequently, in functional analysis, when a linear operator is called "bounded" then it is never meant in t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Young's Inequality For Integral Operators
In mathematical analysis, the Young's inequality for integral operators, is a bound on the L^p\to L^q operator norm of an integral operator in terms of L^r norms of the kernel itself. Statement Assume that X and Y are measurable spaces, K : X \times Y \to \mathbb is measurable and q,p,r\geq 1 are such that \frac = \frac + \frac -1. If : \int_ , K (x, y), ^r \,\mathrm y \le C^r for all x\in X and : \int_ , K (x, y), ^r \,\mathrm x \le C^r for all y\in Y then Theorem 0.3.1 in: C. D. Sogge, ''Fourier integral in classical analysis'', Cambridge University Press, 1993. : \int_ \left, \int_ K (x, y) f(y) \,\mathrm y\^q \, \mathrm x \le C^q \left( \int_ , f(y), ^p \,\mathrm y\right)^\frac. Particular cases Convolution kernel If X = Y = \mathbb^d and K (x, y) = h (x - y) , then the inequality becomes Young's convolution inequality. See also Young's inequality for products In mathematics, Young's inequality for products is a mathematical inequality about the p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Christopher D
Christopher is the English version of a Europe-wide name derived from the Greek name Χριστόφορος (''Christophoros'' or '' Christoforos''). The constituent parts are Χριστός (''Christós''), "Christ" or " Anointed", and φέρειν (''phérein''), "to bear"; hence the "Christ-bearer". As a given name, 'Christopher' has been in use since the 10th century. In English, Christopher may be abbreviated as " Chris", "Topher", and sometimes "Kit". It was frequently the most popular male first name in the United Kingdom, having been in the top twenty in England and Wales from the 1940s until 1995, although it has since dropped out of the top 100. The name is most common in England and not so common in Wales, Scotland, or Ireland. People with the given name Antiquity and Middle Ages * Saint Christopher (died 251), saint venerated by Catholics and Orthodox Christians * Christopher (Domestic of the Schools) (fl. 870s), Byzantine general * Christopher Lekapenos (died 931) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
