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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, generalized functions are objects extending the notion of functions. There is more than one recognized theory, for example the theory of distributions. Generalized functions are especially useful in making
discontinuous function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in valu ...
s more like
smooth function In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if ...
s, and describing discrete physical phenomena such as
point charge A point particle (ideal particle or point-like particle, often spelled pointlike particle) is an idealization of particles heavily used in physics. Its defining feature is that it lacks spatial extension; being dimensionless, it does not take u ...
s. They are applied extensively, especially in
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
and
engineering Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...
. 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 Algebraic analysis is an area of mathematics that deals with systems of linear partial differential equations by using sheaf theory and complex analysis to study properties and generalizations of functions such as hyperfunctions and microfunc ...
. Important influences on the subject have been the technical requirements of theories of
partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to h ...
s, and
group representation In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used ...
theory.


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 mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions. This means that if \operatorname is the linear differenti ...
, in the
Laplace transform In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the '' time domain'') to a function of a complex variable s (in the ...
, and in Riemann's theory of trigonometric series, which were not necessarily the
Fourier series A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or '' ...
of an integrable function. These were disconnected aspects of
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions. These theories are usually studied ...
at the time. The intensive use of the Laplace transform in engineering led to the
heuristic A heuristic (; ), or heuristic technique, is any approach to problem solving or self-discovery that employs a practical method that is not guaranteed to be optimal, perfect, or rational, but is nevertheless sufficient for reaching an immediate ...
use of symbolic methods, called operational calculus. Since justifications were given that used
divergent series 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 finite limit. If a series converges, the individual terms of the series mus ...
, these methods had a bad reputation from the point of view of
pure mathematics Pure mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications, ...
. They are typical of later application of generalized function methods. An influential book on operational calculus was
Oliver Heaviside Oliver Heaviside FRS (; 18 May 1850 – 3 February 1925) was an English self-taught mathematician and physicist who invented a new technique for solving differential equations (equivalent to the Laplace transform), independently develope ...
'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. 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), 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 ...
. That means its value at a given point is (in a sense) not its most important feature. In
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defi ...
a clear formulation is given of the ''essential'' feature of an integrable function, namely the way it defines a
linear functional In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers). If is a vector space over a field , the ...
on other functions. 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 differentiable, but only integrable, i.e., to lie in the L''p'' space L^1( ,b. The method ...
. During the late 1920s and 1930s further steps were taken, basic to future work. The
Dirac delta function In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the enti ...
was boldly defined by
Paul Dirac Paul Adrien Maurice Dirac (; 8 August 1902 – 20 October 1984) was an English theoretical physicist who is regarded as one of the most significant physicists of the 20th century. He was the Lucasian Professor of Mathematics at the Univer ...
(an aspect of his
scientific formalism Scientific formalism is a family of approaches to the presentation of science. It is viewed as an important part of the scientific method, especially in the physical sciences. Levels of formalism There are multiple levels of scientific formalis ...
); this was to treat measures, thought of as densities (such as
charge density In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. Volume charge density (symbolized by the Greek letter ρ) is the quantity of charge per unit volume, measured in the SI system i ...
) like genuine functions.
Sergei Sobolev Prof Sergei Lvovich Sobolev (russian: Серге́й Льво́вич Со́болев) HFRSE (6 October 1908 – 3 January 1989) was a Soviet mathematician working in mathematical analysis and partial differential equations. Sobolev introduced ...
, 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 solution In mathematics, a weak solution (also called a generalized solution) to an ordinary or partial differential equation is a function for which the derivatives may not all exist but which is nonetheless deemed to satisfy the equation in some precise ...
s of partial differential equations. Others proposing related theories at the time were
Salomon Bochner Salomon Bochner (20 August 1899 – 2 May 1982) was an Austrian 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 ...
and Kurt Friedrichs. Sobolev's work was further developed in an extended form by
Laurent Schwartz Laurent-Moïse Schwartz (; 5 March 1915 – 4 July 2002) was a French mathematician. He pioneered the theory of distributions, which gives a well-defined meaning to objects such as the Dirac delta function. He was awarded the Fields Medal in ...
.


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 Laurent-Moïse Schwartz (; 5 March 1915 – 4 July 2002) was a French mathematician. He pioneered the theory of distributions, which gives a well-defined meaning to objects such as the Dirac delta function. He was awarded the Fields Medal in ...
. It can be called a principled theory, based on duality theory for
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
s. Its main rival, in
applied mathematics Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathemati ...
, is to use sequences of smooth approximations (the ' James Lighthill' explanation), which is more ''ad hoc''. This now enters the theory as
mollifier In mathematics, mollifiers (also known as ''approximations to the identity'') are smooth functions with special properties, used for example in distribution theory to create sequences of smooth functions approximating nonsmooth (generalized) f ...
theory. This theory was very successful and is still widely used, but suffers from the main drawback that it allows only
linear Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
operations. In other words, distributions cannot be multiplied (except for very special cases): unlike most classical
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, they are not an
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary ...
. For example it is not meaningful to square the
Dirac delta function In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the enti ...
. Work of Schwartz from around 1954 showed that was an intrinsic difficulty. Some solutions to the multiplication problem have been proposed. One is based on a very simple and intuitive definition a generalized function given by Yu. V. Egorov (see also his article in Demidov's book in the book list below) that allows arbitrary operations on, and between, generalized functions. Another solution of the multiplication problem is dictated by the
path integral formulation The path integral formulation is a description in quantum mechanics that generalizes the action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional i ...
of
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
. Since this is required to be equivalent to the Schrödinger theory of
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
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. The result is equivalent to what can be derived from
dimensional regularization __NOTOC__ In theoretical physics, dimensional regularization is a method introduced by Giambiagi and Bollini as well as – independently and more comprehensively – by 't Hooft and Veltman for regularizing integrals in the evaluation of ...
.


Algebras of generalized functions

Several constructions of algebras of generalized functions have been proposed, among others those by Yu. M. Shirokov and those by E. Rosinger, Y. Egorov, and R. Robinson. 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_ and its singular F_ 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 (); 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. Few applications of the algebra were suggested.


Multiplication of distributions

The problem of ''multiplication of distributions'', a limitation of the Schwartz distribution theory, becomes serious for
non-linear In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...
problems. Various approaches are used today. The simplest one is based on the definition of generalized function given by Yu. V. Egorov. Another approach to construct
associative In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...
differential algebra In mathematics, differential rings, differential fields, and differential algebras are rings, fields, and algebras equipped with finitely many derivations, which are unary functions that are linear and satisfy the Leibniz product rule. A n ...
s is based on J.-F. Colombeau's construction: see
Colombeau algebra In mathematics, a Colombeau algebra is an algebra of a certain kind containing the space of Schwartz distributions. While in classical distribution theory a general multiplication of distributions is not possible, Colombeau algebras provide a rigo ...
. 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.


Example: Colombeau algebra

A simple example is obtained by using the polynomial scale on N, s = \. Then for any semi normed algebra (E,P), the factor space will be :G_s(E,P)= \frac. 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),) (where ''pk'' 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.


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 In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' ...
operation, and :φ''n''(''x'') = ''n'' φ(''nx''). This injection is ''non-canonical ''in the sense that it depends on the choice of the
mollifier In mathematics, mollifiers (also known as ''approximations to the identity'') are smooth functions with special properties, used for example in distribution theory to create sequences of smooth functions approximating nonsmooth (generalized) f ...
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
moment Moment or Moments may refer to: * Present time Music * The Moments, American R&B vocal group Albums * ''Moment'' (Dark Tranquillity album), 2020 * ''Moment'' (Speed album), 1998 * ''Moments'' (Darude album) * ''Moments'' (Christine Guldbrand ...
s up to order ''q'').


Sheaf structure

If (''E'',''P'') is a (pre-) sheaf of semi normed algebras on some topological space ''X'', then ''Gs''(''E'', ''P'') will also have this property. This means that the notion of restriction will be defined, which allows to define the
support Support may refer to: Arts, entertainment, and media * Supporting character Business and finance * Support (technical analysis) * Child support * Customer support * Income Support Construction * Support (structure), or lateral support, a ...
of a generalized function w.r.t. a subsheaf, in particular: * For the subsheaf , one gets the usual support (complement of the largest open subset where the function is zero). * For the subsheaf ''E'' (embedded using the canonical (constant) injection), one gets what is called the singular support, i.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 Lars Valter Hörmander (24 January 1931 – 25 November 2012) was a Swedish mathematician who has been called "the foremost contributor to the modern theory of linear partial differential equations". Hörmander was awarded the Fields Med ...
's '' wave front set'' also for generalized functions. This has an especially important application in the analysis of
propagation Propagation can refer to: *Chain propagation in a chemical reaction mechanism *Crack propagation, the growth of a crack during the fracture of materials * Propaganda, non-objective information used to further an agenda * Reproduction, and other for ...
of singularities.


Other theories

These include: the ''convolution quotient'' theory of Jan Mikusinski, based on the
field of fractions In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field ...
of
convolution In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' ...
algebras that are
integral domain In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural s ...
s; and the theories of hyperfunctions, based (in their initial conception) on boundary values of
analytic function In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
s, and now making use of sheaf theory.


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
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
s that are the typical function domains. The applications are mostly in
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Ma ...
, particularly to
adelic algebraic group In abstract algebra, an adelic algebraic group is a semitopological 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''; t ...
s.
André Weil André Weil (; ; 6 May 1906 – 6 August 1998) was a French mathematician, known for his foundational work in number theory and algebraic geometry. He was a founding member and the ''de facto'' early leader of the mathematical Bourbaki group. Th ...
rewrote Tate's thesis in this language, characterizing the zeta distribution on the
idele group In abstract algebra, an adelic algebraic group is a semitopological 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''; th ...
; and has also applied it to the explicit formula of an 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 that 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 p ...
. 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 real-valued function f is the subset of the function domain containing the elements which are not mapped to zero. If the domain of f is a topological space, then the support of f is instead defined as the smalle ...
. The most developed theory is that of De Rham currents, dual to
differential form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many application ...
s. These are homological in nature, in the way that differential forms give rise to
De Rham cohomology In mathematics, de Rham cohomology (named after Georges de Rham) 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 adap ...
. They can be used to formulate a very general
Stokes' theorem Stokes's theorem, also known as the Kelvin–Stokes theorem Nagayoshi Iwahori, et al.:"Bi-Bun-Seki-Bun-Gaku" Sho-Ka-Bou(jp) 1983/12Written in Japanese)Atsuo Fujimoto;"Vector-Kai-Seki Gendai su-gaku rekucha zu. C(1)" :ja:培風館, Bai-Fu-Kan( ...
.


See also

* Beppo-Levi space *
Dirac delta function In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the enti ...
* Generalized eigenfunction *
Distribution (mathematics) Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives ...
* Hyperfunction * Laplacian of the indicator * Rigged Hilbert space * Limit of a distribution


Books

* L. Schwartz: Théorie des distributions. * A. Beurling, On quasianalyticity and general distributions (Stanford, Calif., 1961). otes by P. L. Duren* I.M. Gel'fand et al.: Generalized Functions, vols I–VI, Academic Press, 1964. (Translated from Russian.) * L. Hörmander: The Analysis of Linear Partial Differential Operators, Springer Verlag, 1983. * H. Komatsu, Introduction to the theory of distributions, Second edition, Iwanami Shoten, Tokyo, 1983. * J.-F. Colombeau: New Generalized Functions and Multiplication of Distributions, North Holland, 1983. * V. S. Vladimirov, Yu. N. Drozhzhinov, and B. I. Zav’yalov, Tauberian theorems for generalized functions, Kluwer Academic Publishers, Dordrecht, 1988. * M. Oberguggenberger: Multiplication of distributions and applications to partial differential equations (Longman, Harlow, 1992). * M. Morimoto, An introduction to Sato’s hyperfunctions, AMS, Providence, RI, 1993. * A. S. Demidov: Generalized Functions in Mathematical Physics: Main Ideas and Concepts (Nova Science Publishers, Huntington, 2001). With an addition by Yu. V. Egorov. * M. Grosser et al.: Geometric theory of generalized functions with applications to general relativity, Kluwer Academic Publishers, 2001. * R. Estrada, R. Kanwal: A distributional approach to asymptotics. Theory and applications, Birkhäuser Boston, Boston, MA, 2002. * V. S. Vladimirov, Methods of the theory of generalized functions, Taylor & Francis, London, 2002. * H. Kleinert, ''Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets'', 4th edition
World Scientific (Singapore, 2006)online here
. See Chapter 11 for products of generalized functions. * S. Pilipovi, B. Stankovic, J. Vindas, Asymptotic behavior of generalized functions, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2012.


References

{{DEFAULTSORT:Generalized Function