In mathematics, generalized functions are objects generalizing the notion of functions. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function 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. In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output In Mathematical analysis, a differentiability class is a classification of functions according to the properties of their Derivatives Higher order differentiability A point charge is an idealized model of a particle which has an Electric charge. They are applied extensively, especially in physics and engineering. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Engineering is the Discipline and Profession of applying technical and scientific Knowledge and

A common feature of some of the approaches is that they build on operator aspects of everyday, numerical functions. In Mathematics, an operator is a function which operates on (or modifies another function 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. Operational calculus is a technique by which problems in analysis in particular differential equations are transformed into algebraic problems usually the problem of solving a polynomial Mikio Sato ( Japanese: 佐藤 幹夫 Sato Mikio; born April 18, 1928) is a Japanese Mathematician, working in what he calls Important influences on the subject have been the technical requirements of theories of partial differential equations, and group representation theory. In Mathematics, partial differential equations ( PDE) are a type of Differential equation, i In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of

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. The 19th century of the Common Era began on January 1, 1801 and ended on December 31, 1900, according to the Gregorian calendar In Mathematics, Green's function is a type of function used to solve inhomogeneous Differential equations subject to boundary conditions In Mathematics, the Laplace transform is one of the best known and most widely used Integral transforms It is commonly used to produce an easily soluble algebraic In Mathematics, a trigonometric series is any series of the form \frac{1}{2}A_{o}+\displaystyle\sum_{n=1}^{\infty}(A_{n} \cos{nx} + B_{n} In Mathematics, a Fourier series decomposes a periodic function into a sum of simple oscillating functions In Mathematics, an integrable function is a function whose Integral exists These were disconnected aspects of mathematical analysis, at the time. 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. heuristic (hyu̇-ˈris-tik is a method to help solve a problem commonly an informal method Operational calculus is a technique by which problems in analysis in particular differential equations are transformed into algebraic problems usually the problem of solving a polynomial Since justifications were given that used divergent series, these methods had a bad reputation from the point of view of pure mathematics. In Mathematics, a divergent series is an Infinite series that is not convergent, meaning that the infinite Sequence of the Partial sums Broadly speaking pure mathematics is Mathematics motivated entirely for reasons other than application They are typical of later application of generalized function methods. An influential book on operational calculus was Oliver Heaviside's Electromagnetic Theory of 1899.

When the Lebesgue integral was introduced, there was for the first time a notion of generalized function central to mathematics. In Mathematics, the Integral of a non-negative function can be regarded in the simplest case as the Area between the graph of An integrable function, in Lebesgue's theory, is equivalent to any other which is the same almost everywhere. 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 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. For functional analysis as used in psychology see the Functional analysis (psychology article This article deals with Linear maps from a Vector space to its field of scalars These maps may be functionals in the traditional This allows a definition of weak derivative. In Mathematics, a weak derivative is a generalization of the concept of the Derivative of a function ( strong derivative) for functions not assumed

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. The Dirac delta or Dirac's delta is a mathematical construct introduced by the British theoretical physicist Paul Dirac. Scientific formalism is a broad term for a family of approaches to the presentation of Science. 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 The linear surface or volume charge density is the amount of Electric charge in a line, Surface, or Volume. 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. Sobolev (masculine and Soboleva (feminine is a popular Russian Surname and may refer to the following people Leonid Sobolev In Mathematics, partial differential equations ( PDE) are a type of Differential equation, i In Mathematics, a weak solution (also called a generalized solution) to an ordinary or Partial differential equation is a function Others proposing related theories at the time were Salomon Bochner and Kurt Friedrichs. Salomon Bochner ( 20 August 1899 &ndash 2 May 1982) was an American Mathematician of Austrian-Hungarian Kurt Otto Friedrichs (1901–1982 was a noted Mathematician. He was the co-founder of the Courant Institute at New York University and recipient of the Sobolev's work was further developed in an extended form by L. Schwartz.

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. In Mathematical analysis, distributions (also known as generalized functions) are objects which generalize functions and Probability distributions Laurent-Moïse Schwartz ( 5 March 1915 in Paris &ndash 4 July 2002 in Paris) was a French Mathematician It can be called a principled theory, based on duality theory for topological vector spaces. In Mathematics, any Vector space V has a corresponding dual vector space (or just dual space for short consisting of all Linear functionals In Mathematics, a topological vector space is one of the basic structures investigated in Functional analysis. Its main rival, in applied mathematics, is to use sequences of smooth approximations (the 'James Lighthill' explanation), which is more ad hoc. 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 This now enters the theory as mollifier theory. In Mathematics, mollifiers (also known as approximations to the identity) are Smooth functions with special properties used in Distribution theory

This theory was very successful and is still widely used, but suffers from the main drawback that it allows only linear operations. The word linear comes from the Latin word linearis, which means created by lines. In other words, distributions cannot be multiplied (except for very special cases): unlike most classical function spaces, they are not an algebra. In Mathematics, a function space is a set of functions of a given kind from a set X to a set Y. Algebra is a branch of Mathematics concerning the study of structure, relation, and Quantity. For example it is not meaningful to square the Dirac delta function. The Dirac delta or Dirac's delta is a mathematical construct introduced by the British theoretical physicist Paul Dirac. Work of Schwartz from around 1954 showed that this was an intrinsic difficulty.

A simple solution of the multiplication problem is dictated by the path integral formulation of quantum mechanics. This article is about a formulation of quantum mechanics For integrals along a path also known as line or contour integrals see Line integral. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons 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. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons This fixes all products of generalized functions as shown by H. Kleinert and A. Hagen Kleinert (born 1941 is Professor of Theoretical Physics at the Free University of Berlin, Germany (since 1968 Honorary Professor at the Kyrgyz-Russian Chervyakov. ^[1] The result is equivalent to what can be derived from dimensional regularization. In Theoretical physics, dimensional regularization is a method for tentatively rendering divergent integrals in the evaluation of Feynman diagrams finite ^[2]

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 to its smooth F_smooth and its singular F_singular parts. The product of generalized functions and 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, . 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. This article describes the use of the term nonlinearity in mathematics Today the most widely used approach to construct such associative differential algebras is based on J. 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, -F. Colombeau's construction: see Colombeau algebra. In Mathematics, the Colombeau algebra is an algebra introduced with the aim of constructing an improved theory of distributions in which multiplication These are factor spaces

G = M / N

of "moderate" modulo "negligible" nets of functions, where "moderateness" and "negligibility" refers to growth with respect to the index of the family. In Topology and related areas of Mathematics, a quotient space (also called an identification space) is intuitively speaking the result of identifying

Example: Colombeau algebra

A simple example is obtained by using the polynomial scale on N, . 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). Non-standard analysis is a branch of Mathematics that formulates analysis using a rigorous notion of an Infinitesimal number 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 Mathematics, the Colombeau algebra is an algebra introduced with the aim of constructing an improved theory of distributions in which multiplication

Injection of Schwartz distributions

This algebra "contains" all distributions T of D' via the injection

j(T) = (φ_n ∗ T)_n + N,

where ∗ is the convolution operation, and

φ_n(x) = n φ(nx). In Mathematics and in particular Functional analysis, convolution is a mathematical operation on two functions f and

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. In Mathematics, mollifiers (also known as approximations to the identity) are Smooth functions with special properties used in Distribution theory 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). In Mathematics, a filter is a special Subset of a Partially ordered set.

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. In Mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the Open sets of a Topological space. This means that the notion of restriction will be defined, which allows to define the support of a generalized function w. In Mathematics, the support of a function is the set of points where the function is not zero or the closure of that set r. t. a subsheaf, in particular:

• For the subsheaf {0}, one gets the usual support (complement of the largest open subset where the function is not zero).
• For the subsheaf E (embedded using the canonical (constant) injection), one gets what is called the singular support, i. In Mathematics, the support of a function is the set of points where the function is not zero or the closure of that set e. , roughly speaking, the closure of the set where the generalized function is not a smooth function (for E=C^∞).

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. This article specifically discusses Fourier transformation of functions on the Real line; for other kinds of Fourier transformation see Fourier analysis and Lars Valter Hörmander (born January 24, 1931 in Mjällby on the peninsula of Listerlandet in the county of Blekinge) is a Swedish In Mathematical analysis, more precisely in Microlocal analysis, the wave front (set WF( f) characterizes the singularities of a Generalized

This has an especially important application in the analysis of propagation of singularities. Wave propagation is any of the ways in which waves travel through a 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

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 Jan Mikusiński ( April 3, 1913 Stanisławów - July 27, 1987 Katowice) was a Polish mathematician known for his In Mathematics, every Integral domain can be embedded in a field; the smallest field which can be used is the field of fractions or field of quotients In Mathematics and in particular Functional analysis, convolution is a mathematical operation on two functions f and In Abstract algebra, a branch of Mathematics, an integral domain is a Commutative ring with an additive identity 0 and a multiplicative identity 1 such In Mathematics, hyperfunctions are generalizations of functions as a 'jump' from one Holomorphic function to another at a boundary and can be thought of informally This article is about both real and complex analytic functions In Mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the Open sets of a Topological space.

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. In Mathematical analysis, distributions (also known as generalized functions) are objects which generalize functions and Probability 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 In Mathematics, a locally compact group is a Topological group G which is locally compact as a Topological space. A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be In Mathematics, the domain of a given function is the set of " Input " values for which the function is defined The applications are mostly in number theory, particularly to adelic algebraic groups. 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 In Mathematics, an adelic algebraic group is a Topological group defined by an Algebraic group G over a Number field K 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. André Weil should not be confused with two other mathematicians with similar names Hermann Weyl (1885-1955 who made substantial contributions In Mathematics, in the field of Number theory, a Hecke character is a generalisation of a Dirichlet character, introduced by Erich Hecke to construct In Mathematics, an adelic algebraic group is a Topological group defined by an Algebraic group G over a Number field K In Mathematics, the explicit formulae for L-functions are a class of summation formulae expressing sums taken over the complex number zeroes of a given L-function

Generalized section

A further way in which the theory has been extended is as generalized sections of a smooth vector bundle. In Mathematics, a vector bundle is a topological construction which makes precise the idea of a family of Vector spaces parameterized by another space This is on the Schwartz pattern, constructing objects dual to the test objects, smooth sections of a bundle that have compact support. In Mathematics, the support of a function is the set of points where the function is not zero or the closure of that set The most developed theory is that of De Rham currents, dual to differential forms. In Mathematics, more particularly in Functional analysis, Differential topology, and Geometric measure theory, a current in the sense of In the mathematical fields of Differential geometry and Tensor calculus, differential forms are an approach to Multivariable calculus which is These are homological in nature, in the way that differential forms give rise to De Rham cohomology. In Mathematics, de Rham cohomology (after Georges de Rham) is a tool belonging both to Algebraic topology and to Differential topology, capable They can be used to formulate a very general Stokes' theorem. In Differential geometry, Stokes' theorem is a statement about the integration of Differential forms which generalizes several Theorems from

See also

• rigged Hilbert space
• generalized eigenfunction
• weak solution

References

Books

• L. Schwartz: Théorie des distributions
• L. Schwartz: Sur l'impossibilité de la multiplication des distributions. Comptes Rendus de L'Academie des Sciences, Paris, 239 (1954) 847-848.
• I.M. Gel'fand et al. Israïl Moiseevich Gelfand (Израиль Моисеевич Гельфанд ישראל געלפֿאַנד (born on) is a Mathematician who has contributed substantially : Generalized Functions, vols I–VI, Academic Press, 1964–. (Translated from Russian. )
• L. Hörmander: The Analysis of Linear Partial Differential Operators, Springer Verlag, 1983.
• J. -F. Colombeau: New Generalized Functions and Multiplication of the Distributions, North Holland, 1983.
• M. Grosser et al. : Geometric theory of generalized functions with applications to general relativity, Kluwer Academic Publishers, 2001.
• H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 4th edition, World Scientific (Singapore, 2006)(also available online here). Hagen Kleinert (born 1941 is Professor of Theoretical Physics at the Free University of Berlin, Germany (since 1968 Honorary Professor at the Kyrgyz-Russian See Chapter 11 for products of generalized functions.