NationMaster - Encyclopedia: Generalized function

In mathematics, generalized functions are objects generalizing the notion of functions. There is more than one recognised theory. Generalized functions are especially useful in making discontinuous functions more like smooth functions, and (going to extremes) describing physical phenomena such as point charges. They are applied extensively, especially in physics and engineering. In certain systems for object-oriented programming such as the Common Lisp Object System and Dylan, a generic function is an entity made up of all methods having the same name. ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... This article is about functions in mathematics. ... In mathematics, a continuous function is one in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ... In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. ... A point charge is an idealized model of a particle which has an electric charge. ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... Engineering is the discipline and profession of applying scientific knowledge and utilizing natural laws and physical resources in order to design and implement materials, structures, machines, devices, systems, and processes that realize a desired objective and meet specified criteria. ...

A common feature of some of the approaches is that they build on operator aspects of everyday, numerical functions. The early history is connected with some ideas on operational calculus, and more contemporary developments in certain directions are closely related to ideas of Mikio Sato, on what he calls algebraic analysis. Important influences on the subject have been the technical requirements of theories of partial differential equations, and group representation theory. In mathematics, an operator is a function that performs some sort of operation on a number, variable, or function. ... Mikio Sato (佐藤幹夫, born April 18, 1928) is a Japanese mathematician, working in what he calls algebraic analysis. ... In mathematics, a partial differential equation (PDE) is a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables. ... Group representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. ...

Some early history

In the mathematics of the nineteenth century, aspects of generalized function theory appeared, for example in the definition of the Green's function, in the Laplace transform, and in Riemann's theory of trigonometric series, which were not necessarily the Fourier series of an integrable function. These were disconnected aspects of mathematical analysis, at the time. Alternative meaning: Nineteenth Century (periodical) (18th century — 19th century — 20th century — more centuries) As a means of recording the passage of time, the 19th century was that century which lasted from 1801-1900 in the sense of the Gregorian calendar. ... In mathematics, a Greens function is a type of function used to solve inhomogeneous differential equations subject to boundary conditions. ... In the branch of mathematics called functional analysis, the Laplace transform, , is a linear operator on a function f(t) (original) with a real argument t (t â‰¥ 0) that transforms it to a function F(s) (image) with a complex argument s. ... Bernhard Riemann. ... In mathematics, a Fourier series of a periodic function, named in honor of Joseph Fourier (1768-1830), represents the function as a sum of periodic functions of the form where e is Eulers number and i the imaginary unit. ... The Fourier series is a mathematical tool used for analyzing periodic functions by decomposing such a function into a weighted sum of much simpler sinusoidal component functions sometimes referred to as normal Fourier modes, or simply modes for short. ... In mathematics, the term integrable function refers to a function whose integral may be calculated. ... Analysis has its beginnings in the rigorous formulation of calculus. ...

The intensive use of the Laplace transform in engineering led to the heuristic use of symbolic methods, called operational calculus. Since justifications were given that used divergent series, these methods had a bad reputation from the point of view of pure mathematics. They are typical of later application of generalized function methods. An influential book on operational calculus was Oliver Heaviside's Electromagnetic Theory of 1899. For other uses, see Heuristic (disambiguation). ... In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a limit. ... Broadly speaking, pure mathematics is mathematics motivated entirely for reasons other than application. ... Oliver Heaviside (May 18, 1850 â€“ February 3, 1925) was a self-taught English electrical engineer, mathematician, and physicist who adapted complex numbers to the study of electrical circuits, developed techniques for applying Laplace transforms to the solution of differential equations, reformulated Maxwells field equations in terms of electric and...

When the Lebesgue integral was introduced, there was for the first time a notion of generalized function central to mathematics. An integrable function, in Lebesgue's theory, is equivalent to any other which is the same almost everywhere. That means its value at a given point is (in a sense) not its most important feature. In functional analysis a clear formulation is given of the essential feature of an integrable function, namely the way it defines a linear functional on other functions. This allows a definition of weak derivative. In mathematics, the integral of a function of one real variable can be regarded as the area of a plane region bounded by the graph of that function. ... In measure theory (a branch of mathematical analysis), one says that a property holds almost everywhere if the set of elements for which the property does not hold is a null set, i. ... Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. ... In linear algebra, a branch of mathematics, a linear functional or linear form is a linear function from a vector space to its field of scalars. ... In mathematics, a weak derivative is a generalization of the concept of the derivative of a function (strong derivative) for functions not assumed differentiable, but only integrable, i. ...

During the late 1920s and 1930s further steps were taken, basic to future work. The Dirac delta function was boldly defined by Paul Dirac (an aspect of his scientific formalism); this was to treat measures, thought of as densities (such as charge density) like honest functions. Sobolev, working in partial differential equation theory, defined the first adequate theory of generalized functions, from the mathematical point of view, in order to work with weak solutions of PDEs. Others proposing related theories at the time were Salomon Bochner and Kurt Friedrichs. Sobolev's work was further developed in an extended form by L. Schwartz. The Dirac delta or Diracs delta, often referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function Î´(x) that has the value of infinity for x = 0 and the value zero elsewhere. ... Paul Adrien Maurice Dirac, OM, FRS (IPA: [dÉªrÃ¦k]) (August 8, 1902 â€“ October 20, 1984) was a British theoretical physicist and a founder of the field of quantum physics. ... Scientific formalism is a possible term for two aspects of the presentation of science, particularly relevant to the physical sciences. ... In mathematics the concept of a measure generalizes notions such as length, area, and volume (but not all of its applications have to do with physical sizes). ... Charge density is the amount of electric charge per unit volume. ... Sobolev, Sergei Lvovich (Russian: Сергей Львович Соболев) (6 October 1908- 3 January 1989) was a Russian mathematician, working in mathematical analysis and partial differential equations. ... In mathematics, and in particular analysis, a partial differential equation (PDE) is an equation involving partial derivatives of an unknown function. ... In mathematics, a weak solution (also called a generalized solution) to an ordinary or partial differential equation is a function for which the derivatives appearing in the equation may not all exist but which is nonetheless deemed to satisfy the equation in some precisely defined sense. ... Salomon Bochner (20 August 1899 - 2 May 1982) was a Polish-American mathematician, known for wide-ranging work in mathematical analysis, probability theory and differential geometry. ... Kurt O. Friedrichs (1901-1982) was a noted mathematician. ...

Schwartz distributions

The realization of such a concept that was to become accepted as definitive, for many purposes, was the theory of distributions, developed by Laurent Schwartz. It can be called a principled theory, based on duality theory for topological vector spaces. Its main rival, in applied mathematics, is to use sequences of smooth approximations (the 'James Lighthill' explanation), which is more ad hoc. This now enters the theory as mollifier theory. This article is about generalized functions in mathematical analysis. ... Laurent Schwartz (5 March 1915 â€“ 4 July 2002 in Paris) was a French mathematician. ... In mathematics, any vector space V has a corresponding dual vector space (or just dual space for short) consisting of all linear functionals on V. Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors which are studied in tensor algebra. ... In mathematics a topological vector space is one of the basic structures investigated in functional analysis. ... Applied mathematics is a branch of mathematics that concerns itself with the mathematical techniques typically used in the application of mathematical knowledge to other domains. ... Sir Michael James Lighthill FRS (23 January 1924 - 17 July 1998) was a British applied mathematician, known for his pioneering work in the field of Aeroacoustics. ... A mollifier (top). ...

This theory was very successful and is still widely used, but suffers from the main drawback that it allows only linear operations. In other words, distributions cannot be multiplied (except for very special cases): unlike most classical function spaces, they are not an algebra. For example it is not meaningful to square the Dirac delta function. Work of Schwartz from around 1954 showed that this was an intrinsic difficulty. For other uses, see Linear (disambiguation). ... In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in many applications, it is a topological space or a vector space or both. ... This article is about the branch of mathematics. ... The Dirac delta or Diracs delta, often referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function Î´(x) that has the value of infinity for x = 0 and the value zero elsewhere. ...

A simple solution of the multiplication problem is dictated by the path integral formulation of quantum mechanics. Since this is required to be equivalent to the Schrödinger theory of quantum mechanics which is invariant under coordinate transformations, this property must be shared by path integrals. This fixes all products of generalized functions as shown by H. Kleinert and A. Chervyakov.^[1] The result is equivalent to what can be derived from dimensional regularization.^[2] This article or section is in need of attention from an expert on the subject. ... For a generally accessible and less technical introduction to the topic, see Introduction to quantum mechanics. ... Erwin SchrÃ¶dinger, as depicted on the former Austrian 1000 Schilling bank note. ... For a generally accessible and less technical introduction to the topic, see Introduction to quantum mechanics. ... Hagen Kleinert, Photo taken in 2006 Hagen Kleinert is Professor of Theoretical Physics at the Free University of Berlin, Germany (since 1968), Honorary Professor at the Kyrgyz-Russian Slavic University, and Honorary Member of the Russian Academy of Creative Endeavors. ... In theoretical physics, dimensional regularization is a particular way to get rid of infinities that occur when one evaluates Feynman diagrams in quantum field theory. ...

Algebras of generalized functions

Several constructions of algebras of generalized functions have been proposed, among others those by Yu.M.Shirokov ^[3] and those by E. Rosinger, Y. Egorov, and R. Robinson ^[4]. In the first case, the multiplication is determined with some regularization of generalized function. In the second case, the algebra is constructed as multiplication of distributions. Both cases are discussed below.

Non-commutative algebra of generalized functions

The algebra of generalized functions can be built-up with an appropriate procedure of projection of a function $~F=F(x)~$ to its smooth

F smooth

and its singular

F singular

parts. The product of generalized functions $~F~$ and $~G~$ appears as

Such a rule applies to both, the space of main functions and the space of operators which act on the space of the main functions. The associativity of multiplication is achieved; and the function signum is defined in such a way, that its square is unity everywhere (including the origin of coordinates). Note that the product of singular parts does not appear in the right-hand side of (1); in particular, $~delta(x)^2=0~$ . Such a formalism includes the conventional theory of generalized functions (without their product) as a special case. However, the resulting algebra is non-commutative: generalized functions signum and delta anticommute ^[3]. Few applications of the algebra were suggested ^[5] ^[6].

Multiplication of distributions

The problem of multiplication of distributions, a limitation of the Schwartz distribution theory becomes serious for non-linear problems. Today the most widely used approach to construct such associative differential algebras is based on J.-F. Colombeau's construction: see Colombeau algebra. These are factor spaces To do: 20th century mathematics chaos theory, fractals Lyapunov stability and non-linear control systems non-linear video editing See also: Aleksandr Mikhailovich Lyapunov Dynamical system External links http://www. ... In mathematics, associativity is a property that a binary operation can have. ... In mathematics, differential rings, differential fields and differential algebras are rings, fields and algebras equipped with a derivation, which is a unary function satisfying the Leibniz product law. ... In mathematics, the Colombeau algebra is an algebra introduced with the aim of constructing an improved theory of distributions, in which multiplication is not problematic. ... In topology and related areas of mathematics, a quotient space (also called an identification space) is, intuitively speaking, the result of identifying or gluing together certain points of a given space. ...

of "moderate" modulo "negligible" nets of functions, where "moderateness" and "negligibility" refers to growth with respect to the index of the family.

Example: Colombeau algebra

A simple example is obtained by using the polynomial scale on N, $s = { a_m:mathbb Ntomathbb R, nmapsto n^m ;~ minmathbb Z }$ . Then for any semi normed algebra (E,P), the factor space will be

In particular, for (E, P)=(C,|.|) one gets (Colombeau's) generalized complex numbers (which can be "infinitely large" and "infinitesimally small" and still allow for rigorous arithmetics, very similar to nonstandard numbers). For (E, P) = (C^∞(R),{p_k}) (where p_k is the supremum of all derivatives of order less than or equal to k on the ball of radius k) one gets Colombeau's simplified algebra. In the most restricted sense, nonstandard analysis or non-standard analysis is that branch of mathematics that formulates analysis using a rigorous notion of infinitesimal, where an element of an ordered field F is infinitesimal if and only if its absolute value is smaller than any element of F of... In mathematics, the Colombeau algebra is an algebra introduced with the aim of constructing an improved theory of distributions, in which multiplication is not problematic. ...

Injection of Schwartz distributions

where ∗ is the convolution operation, and In mathematics and, in particular, functional analysis, convolution is a mathematical operator which takes two functions f and g and produces a third function that in a sense represents the amount of overlap between f and a reversed and translated version of g. ...

This injection is non-canonical in the sense that it depends on the choice of the mollifier φ, which should be C^∞, of integral one and have all its derivatives at 0 vanishing. To obtain a canonical injection, the indexing set can be modified to be N × D(R), with a convenient filter base on D(R) (functions of vanishing moments up to order q). A mollifier (top). ... In mathematics, a filter is a special subset of a partially ordered set. ... -1...

Sheaf structure

If (E,P) is a (pre-)sheaf of semi normed algebras on some topological space X, then G_s(E,P) will also have this property. This means that the notion of restriction will be defined, which allows to define the support of a generalized function w.r.t. a subsheaf, in particular: In mathematics, a sheaf F on a topological space X is something that assigns a structure F(U) (such as a set, group, or ring) to each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and... In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ... In mathematics, the support of a real-valued function f on a set X is sometimes defined as the subset of X on which f is nonzero. ...

In mathematics, the support of a numerical function f on a set X is sometimes defined as the subset of X on which f is nonzero. ...

Microlocal analysis

The Fourier transformation being (well-)defined for compactly supported generalized functions (component-wise), one can apply the same construction as for distributions, and define Lars Hörmander's wave front set also for generalized functions. The Fourier transform, named after Jean Baptiste Joseph Fourier, is an integral transform that re-expresses a function in terms of sinusoidal basis functions, i. ... Lars HÃ¶rmander Lars Valter HÃ¶rmander (born 24 January 1931) is a Swedish mathematician and one of the leading experts in partial differential equations. ... In mathematical analysis, more precisely in microlocal analysis, the wave front (set) WF(f) characterizes the singularities of a generalized function f, not only in space, but more precisely also with respect to its Fourier transform at each point. ...

This has an especially important application in the analysis of propagation of singularities. Wave propagation refers to the ways waves travel through a medium (waveguide). ... In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability. ...

Other theories

These include: the convolution quotient theory of Jan Mikusinski , based on the field of fractions of convolution algebras that are integral domains; and the theories of hyperfunctions, based (in their initial conception) on boundary values of analytic functions, and now making use of sheaf theory. Prof. ... In mathematics, every integral domain can be embedded in a field; the smallest field which can be used is the field of fractions of the integral domain. ... In mathematics and, in particular, functional analysis, convolution is a mathematical operator which takes two functions f and g and produces a third function that in a sense represents the amount of overlap between f and a reversed and translated version of g. ... In abstract algebra, an integral domain is a commutative ring with an additive identity 0 and a multiplicative identity 1 such that 0 â‰ 1, in which the product of any two non-zero elements is always non-zero; that is, there are no zero divisors. ... In mathematics, hyperfunctions are sums of boundary values of holomorphic functions, and can be thought of informally as distributions of infinite order. ... Not to be confused with Analytic signal. ... In mathematics, a sheaf F on a given topological space X gives a set or richer structure F(U) for each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and gluing smaller open sets to obtain...

Topological groups

Bruhat introduced a class of test functions, the Schwartz-Bruhat functions as they are now known, on a class of locally compact groups that goes beyond the manifolds that are the typical function domains. The applications are mostly in number theory, particularly to adelic algebraic groups. André Weil rewrote Tate's thesis in this language, characterising the zeta distribution on the idele group; and has also applied it to the explicit formula of an L-function. This page deals with mathematical distributions. ... In mathematics, a Schwartz-Bruhat function is a function on a locally compact abelian group, such as the adeles, that generalizes a Schwartz function on a real vector space. ... In mathematics, a topological group G is a group that is also a topological space such that the group multiplication G × G → G and the inverse operation G → G are continuous maps. ... On a sphere, the sum of the angles of a triangle is not equal to 180Â° (see spherical trigonometry). ... In mathematics, the domain of a function is the set of all input values to the function. ... Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ... In mathematics, an adelic algebraic group is a topological group defined by an algebraic group G over a number field K, and the adele ring A = A(K) of K. It consists of the points of G having values in A; the definition of the appropriate topology is straightforward only... AndrÃ© Weil (May 6, 1906 - August 6, 1998) (pronounced [1]) was one of the greatest mathematicians of the 20th century, whether measured by his research work, its influence on future work, exposition or breadth. ... In mathematics, an adelic algebraic group is a topological group defined by an algebraic group G over a number field K, and the adele ring A = A(K) of K. It consists of the points of G having values in A; the definition of the appropriate topology is straightforward only...

Generalized section

A further way in which the theory has been extended is as generalized sections of a smooth vector bundle. This is on the Schwartz pattern, constructing objects dual to the test objects, smooth sections of a bundle that have compact support. The most developed theory is that of De Rham currents, dual to differential forms. These are homological in nature, in the way that differential forms give rise to De Rham cohomology. They can be used to formulate a very general Stokes' theorem. In mathematics, a vector bundle is a topological construction which makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every point x of the space X we associate (or... In mathematics, the support of a numerical function f on a set X is sometimes defined as the subset of X on which f is nonzero. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... In mathematics, de Rham cohomology is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes. ... Stokes theorem in differential geometry is a statement about the integration of differential forms which generalizes several theorems from vector calculus. ...

Contents