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Harish-Chandra's Schwartz Space
In mathematical abstract harmonic analysis, Harish-Chandra's Schwartz space is a space of functions on a semisimple Lie group whose derivatives are rapidly decreasing, studied by . It is an analogue of the Schwartz space on a real vector space, and is used to define the space of tempered distributions on a semisimple Lie group. Definition The definition of the Schwartz space uses Harish-Chandra's Ξ function and his ''σ'' function. The ''σ'' function is defined by : \sigma(x)=\, X\, for ''x''=''k'' exp ''X'' with ''k'' in ''K'' and ''X'' in ''p'' for a Cartan decomposition ''G'' = ''K'' exp ''p'' of the Lie group ''G'', where , , ''X'', , is a ''K''-invariant Euclidean norm on ''p'', usually chosen to be the Killing form. . The Schwartz space on ''G'' consists roughly of the functions all of whose derivatives are rapidly decreasing compared to ''Ξ''. More precisely, if ''G'' is connected then the Schwartz space consists of all smooth funct ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harmonic Analysis
Harmonic analysis is a branch of mathematics concerned with the representation of Function (mathematics), functions or signals as the Superposition principle, superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms (i.e. an extended form of Fourier analysis). In the past two centuries, it has become a vast subject with applications in areas as diverse as number theory, representation theory, signal processing, quantum mechanics, tidal analysis and neuroscience. The term "harmonics" originated as the Ancient Greek word ''harmonikos'', meaning "skilled in music". In physical eigenvalue problems, it began to mean waves whose frequencies are Multiple (mathematics), integer multiples of one another, as are the frequencies of the Harmonic series (music), harmonics of music notes, but the term has been generalized beyond its original meaning. The classical Fourier transform on R''n'' is still an area of ongoing research, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semisimple Lie Group
In mathematics, a Lie algebra is semisimple if it is a direct sum of modules, direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper Lie algebra#Subalgebras.2C ideals and homomorphisms, ideals). Throughout the article, unless otherwise stated, a Lie algebra is a finite-dimensional Lie algebra over a field of Characteristic (algebra), characteristic 0. For such a Lie algebra \mathfrak g, if nonzero, the following conditions are equivalent: *\mathfrak g is semisimple; *the Killing form, κ(x,y) = tr(ad(''x'')ad(''y'')), is non-degenerate; *\mathfrak g has no non-zero abelian ideals; *\mathfrak g has no non-zero solvable Lie algebra, solvable ideals; * the Radical of a Lie algebra, radical (maximal solvable ideal) of \mathfrak g is zero. Significance The significance of semisimplicity comes firstly from the Levi decomposition, which states that every finite dimensional Lie algebra is the semidirect product of a solvable i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schwartz Space
In mathematics, Schwartz space \mathcal is the function space of all Function (mathematics), functions whose derivatives are rapidly decreasing. This space has the important property that the Fourier transform is an automorphism on this space. This property enables one, by duality, to define the Fourier transform for elements in the dual space \mathcal^* of \mathcal, that is, for tempered distributions. A function in the Schwartz space is sometimes called a Schwartz function. Schwartz space is named after French mathematician Laurent Schwartz. Definition Let \mathbb be the Set (mathematics), set of non-negative integers, and for any n \in \mathbb, let \mathbb^n := \underbrace_ be the ''n''-fold Cartesian product. The ''Schwartz space'' or space of rapidly decreasing functions on \mathbb^n is the function spaceS \left(\mathbb^n, \mathbb\right) := \left \,where C^(\mathbb^n, \mathbb) is the function space of smooth functions from \mathbb^n into \mathbb, and\, f\, _:= \sup_ \lef ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can be complex numbers or, more generally, elements of any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called ''vector axioms''. The terms real vector space and complex vector space are often used to specify the nature of the scalars: real coordinate space or complex coordinate space. Vector spaces generalize Euclidean vectors, which allow modeling of physical quantities, such as forces and velocity, that have not only a magnitude, but also a direction. The concept of vector spaces is fundamental for linear algebra, together with the concept of matrix, which allows computing in vector spaces. This provides a concise and synthetic way for manipulating and studying systems of linear eq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tempered Representation
In mathematics, a tempered representation of a linear semisimple Lie group is a representation that has a basis whose matrix coefficients lie in the L''p'' space :''L''2+ε(''G'') for any ε > 0. Formulation This condition, as just given, is slightly weaker than the condition that the matrix coefficients are square-integrable, in other words lie in :''L''2(''G''), which would be the definition of a discrete series representation. If ''G'' is a linear semisimple Lie group with a maximal compact subgroup ''K'', an admissible representation ρ of ''G'' is tempered if the above condition holds for the ''K''-finite matrix coefficients of ρ. The definition above is also used for more general groups, such as ''p''-adic Lie groups and finite central extensions of semisimple real algebraic groups. The definition of "tempered representation" makes sense for arbitrary unimodular locally compact groups, but on groups with infinite centers such as infinite central extensions ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harish-Chandra's Ξ Function
In mathematical harmonic analysis, Harish-Chandra's ''Ξ'' function is a special spherical function on a semisimple Lie group, studied by . Harish-Chandra used it to define Harish-Chandra's Schwartz space. gives a detailed description of the properties of ''Ξ''. Definition :\Xi(g)=\int_Ka(kg)^\rho dk, where *''K'' is a maximal compact subgroup of a semisimple Lie group with Iwasawa decomposition ''G''=''NAK'' *''g'' is an element of ''G'' *''ρ'' is a Weyl vector In mathematics, the Weyl character formula in representation theory describes the characters of irreducible representations of compact Lie groups in terms of their highest weights. It was proved by . There is a closely related formula for the c ... *''a''(''g'') is the element ''a'' in the Iwasawa decomposition ''g''=''nak'' References * * {{DEFAULTSORT:Harish-Chandra's Xi function Harmonic analysis Representation theory Special functions ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartan Decomposition
In mathematics, the Cartan decomposition is a decomposition of a Semisimple Lie algebra, semisimple Lie group or Lie algebra, which plays an important role in their structure theory and representation theory. It generalizes the polar decomposition or singular value decomposition of matrices. Its history can be traced to the 1880s work of Élie Cartan and Wilhelm Killing. Cartan involutions on Lie algebras Let \mathfrak be a real semisimple Lie algebra and let B(\cdot,\cdot) be its Killing form. An Involution (mathematics), involution on \mathfrak is a Lie algebra automorphism \theta of \mathfrak whose square is equal to the identity. Such an involution is called a ''Cartan involution'' on \mathfrak if B_\theta(X,Y) := -B(X,\theta Y) is a positive definite bilinear form. Two involutions \theta_1 and \theta_2 are considered equivalent if they differ only by an inner automorphism. Any real semisimple Lie algebra has a Cartan involution, and any two Cartan involutions are equi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Killing Form
In mathematics, the Killing form, named after Wilhelm Killing, is a symmetric bilinear form that plays a basic role in the theories of Lie groups and Lie algebras. Cartan's criteria (criterion of solvability and criterion of semisimplicity) show that Killing form has a close relationship to the semisimplicity of the Lie algebras. History and name The Killing form was essentially introduced into Lie algebra theory by in his thesis. In a historical survey of Lie theory, has described how the term ''"Killing form"'' first occurred in 1951 during one of his own reports for the Séminaire Bourbaki; it arose as a misnomer, since the form had previously been used by Lie theorists, without a name attached. Some other authors now employ the term ''" Cartan-Killing form"''. At the end of the 19th century, Killing had noted that the coefficients of the characteristic equation of a regular semisimple element of a Lie algebra are invariant under the adjoint group, from which it follows tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Acta Mathematica
''Acta Mathematica'' is a peer-reviewed open-access scientific journal covering research in all fields of mathematics. According to Cédric Villani, this journal is "considered by many to be the most prestigious of all mathematical research journals".. According to the ''Journal Citation Reports'', the journal has a 2020 impact factor of 4.273, ranking it 5th out of 330 journals in the category "Mathematics". Publication history The journal was established by Gösta Mittag-Leffler in 1882 and is published by Institut Mittag-Leffler, a research institute for mathematics belonging to the Royal Swedish Academy of Sciences. The journal was printed and distributed by Springer from 2006 to 2016. Since 2017, Acta Mathematica has been published electronically and in print by International Press. Its electronic version is open access without publishing fees. Poincaré episode The journal's "most famous episode" (according to Villani) concerns Henri Poincaré, who won a prize offered ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Academic Press
Academic Press (AP) is an academic book publisher founded in 1941. It was acquired by Harcourt, Brace & World in 1969. Reed Elsevier bought Harcourt in 2000, and Academic Press is now an imprint of Elsevier. Academic Press publishes reference books, serials and online products in the subject areas of: * Communications engineering * Economics * Environmental science * Finance * Food science and nutrition * Geophysics * Life sciences * Mathematics and statistics * Neuroscience * Physical sciences * Psychology Well-known products include the ''Methods in Enzymology'' series and encyclopedias such as ''The International Encyclopedia of Public Health'' and the ''Encyclopedia of Neuroscience''. See also * Akademische Verlagsgesellschaft (AVG) — the German predecessor, founded in 1906 by Leo Jolowicz (1868–1940), the father of Walter Jolowicz Walter may refer to: People * Walter (name), both a surname and a given name * Little Walter, American blues harmonica player Marion Wa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harmonic Analysis
Harmonic analysis is a branch of mathematics concerned with the representation of Function (mathematics), functions or signals as the Superposition principle, superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms (i.e. an extended form of Fourier analysis). In the past two centuries, it has become a vast subject with applications in areas as diverse as number theory, representation theory, signal processing, quantum mechanics, tidal analysis and neuroscience. The term "harmonics" originated as the Ancient Greek word ''harmonikos'', meaning "skilled in music". In physical eigenvalue problems, it began to mean waves whose frequencies are Multiple (mathematics), integer multiples of one another, as are the frequencies of the Harmonic series (music), harmonics of music notes, but the term has been generalized beyond its original meaning. The classical Fourier transform on R''n'' is still an area of ongoing research, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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