# Generalized function

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{{Short description|Objects extending the notion of functions}}
In [mathematics](/source/mathematics), '''generalized functions''' are objects extending the notion of [function](/source/function_(mathematics))s on real or complex numbers. There is more than one recognized theory, for example  the theory of [distributions](/source/distribution_(mathematics)). Generalized functions are especially useful for treating [discontinuous function](/source/discontinuous_function)s more like [smooth function](/source/smooth_function)s, and describing discrete physical phenomena such as [point charge](/source/point_charge)s. They are applied extensively, especially in [physics](/source/physics) and [engineering](/source/engineering). Important motivations have been the technical requirements of theories of [partial differential equation](/source/partial_differential_equation)s and [group representations](/source/group_representation).

A common feature of some of the approaches is that they build on [operator](/source/Operator_(mathematics)) aspects of everyday, numerical functions. The early history is connected with some ideas on [operational calculus](/source/operational_calculus), and some contemporary developments are closely related to [Mikio Sato](/source/Mikio_Sato)'s [algebraic analysis](/source/algebraic_analysis). 

==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](/source/Green's_function), in the [Laplace transform](/source/Laplace_transform), and in [Riemann](/source/Riemann)'s theory of [trigonometric series](/source/trigonometric_series), which were not necessarily the [Fourier series](/source/Fourier_series) of an [integrable function](/source/integrable_function). These were disconnected aspects of [mathematical analysis](/source/mathematical_analysis) at the time.

The intensive use of the Laplace transform in engineering led to the [heuristic](/source/heuristic) use of symbolic methods, called [operational calculus](/source/operational_calculus). Since justifications were given that used [divergent series](/source/divergent_series), these methods were questionable from the point of view of [pure mathematics](/source/pure_mathematics). They are typical of later application of generalized function methods. An influential book on operational calculus was [Oliver Heaviside](/source/Oliver_Heaviside)'s ''Electromagnetic Theory'' of 1899.

When the [Lebesgue integral](/source/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](/source/almost_everywhere). That means its value at each point is (in a sense) not its most important feature. In [functional analysis](/source/functional_analysis) a clear formulation is given of the ''essential'' feature of an integrable function, namely the way it defines a [linear functional](/source/linear_functional) on other functions. This allows a definition of [weak derivative](/source/weak_derivative).

During the late 1920s and 1930s further basic steps were taken. The [Dirac delta function](/source/Dirac_delta_function) was boldly defined by [Paul Dirac](/source/Paul_Dirac) (an aspect of his [scientific formalism](/source/scientific_formalism)); this was to treat [measures](/source/measure_(mathematics)), thought of as densities (such as [charge density](/source/charge_density)) like genuine functions. [Sergei Sobolev](/source/Sergei_Sobolev), working in [partial differential equation theory](/source/partial_differential_equation_theory), defined the first rigorous theory of generalized functions in order to define [weak solution](/source/weak_solution)s of partial differential equations (i.e. solutions which are generalized functions, but may not be ordinary functions).<ref>{{Cite book |last1=Kolmogorov |first1=A. N. |title=Elements of the theory of functions and functional analysis |last2=Fomin |first2=S. V. |author-link=Andrey Kolmogorov|author-link2=Sergei Fomin |date=1999 |publisher=Dover |orig-date=1957 |isbn=0-486-40683-0 |location=Mineola, N.Y. |oclc=44675353}}</ref> Others proposing related theories at the time were [Salomon Bochner](/source/Salomon_Bochner) and [Kurt Friedrichs](/source/Kurt_Friedrichs). Sobolev's work was extended by [Laurent Schwartz](/source/Laurent_Schwartz).<ref>{{cite journal | last1 = Schwartz | first1 = L | year = 1952 | title = Théorie des distributions | journal = Bull. Amer. Math. Soc. | volume = 58 | pages = 78–85 | doi = 10.1090/S0002-9904-1952-09555-0 | doi-access = free }}</ref>

==Schwartz distributions==

The most definitive development was the theory of [distributions](/source/distribution_(mathematics)) developed by [Laurent Schwartz](/source/Laurent_Schwartz), systematically working out the principle of [duality](/source/dual_space) for [topological vector space](/source/topological_vector_space)s. Its main rival in [applied mathematics](/source/applied_mathematics) is [mollifier](/source/mollifier) theory, which uses sequences of smooth approximations (the '[James Lighthill](/source/James_Lighthill)' explanation).<ref>Halperin, I., & Schwartz, L. (1952). Introduction to the Theory of Distributions. Toronto: University of Toronto Press. (Short lecture by Halperin on Schwartz's theory)</ref>

This theory was very successful and is still widely used, but suffers from the main drawback that distributions cannot usually be multiplied: unlike most classical [function space](/source/function_space)s, they do not form an [algebra](/source/algebra). For example, it is meaningless to square the [Dirac delta function](/source/Dirac_delta_function). Work of Schwartz from around 1954 showed this to be an intrinsic difficulty.

==Algebras of generalized functions==

Some solutions to the multiplication problem have been proposed. One is based on a simple definition of by Yu. V. Egorov<ref name="YuVEgorov1990">
{{cite journal |author=Yu. V. Egorov |year=1990 |title=A contribution to the theory of generalized functions |journal=Russian Math. Surveys |volume=45 |issue=5 |pages=1–49 |bibcode=1990RuMaS..45....1E |doi=10.1070/rm1990v045n05abeh002683 |s2cid=250877163}}</ref> (see also his article in Demidov's book in the book list below) that allows arbitrary operations on, and between, generalized functions.

Another solution allowing multiplication is suggested by the [path integral formulation](/source/path_integral_formulation) of [quantum mechanics](/source/quantum_mechanics).
Since this is required to be equivalent to the [Schrödinger](/source/Schr%C3%B6dinger) theory of [quantum mechanics](/source/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](/source/Hagen_Kleinert) and A. Chervyakov.<ref>{{cite journal |author=H. Kleinert and A. Chervyakov |year=2001 |title=Rules for integrals over products of distributions from coordinate independence of path integrals |url=http://www.physik.fu-berlin.de/~kleinert/kleiner_re303/wardepl.pdf |journal=Eur. Phys. J. C |volume=19 |issue=4 |pages=743–747 |arxiv=quant-ph/0002067 |bibcode=2001EPJC...19..743K |doi=10.1007/s100520100600 |s2cid=119091100 |archive-date=2008-04-08 |access-date=2007-10-05 |archive-url=https://web.archive.org/web/20080408123949/http://www.physik.fu-berlin.de/~kleinert/kleiner_re303/wardepl.pdf |url-status=dead }}</ref> The result is equivalent to what can be derived from
[dimensional regularization](/source/dimensional_regularization).<ref>
{{cite journal |author=H. Kleinert and A. Chervyakov |year=2000 |title=Coordinate Independence of Quantum-Mechanical Path Integrals |url=http://www.physik.fu-berlin.de/~kleinert/305/klch2.pdf |journal=Phys. Lett. |volume=A 269 |issue=1–2 |page=63 |arxiv=quant-ph/0003095 |bibcode=2000PhLA..273....1K |doi=10.1016/S0375-9601(00)00475-8}}</ref>

Several constructions of algebras of generalized functions have been proposed, among others those by Yu. M. Shirokov
<ref name="shirokovAlgebra1dim">{{cite journal
|author=Yu. M. Shirokov
|title=Algebra of one-dimensional generalized functions
|journal=[Theoretical and Mathematical Physics](/source/Theoretical_and_Mathematical_Physics)
|year=1979
|volume=39
|issue=3
|pages=291–301
|url=http://en.wikisource.org/wiki/Algebra_of_generalized_functions_%28Shirokov%29
|bibcode=1979TMP....39..471S
|doi=10.1007/BF01017992
|s2cid=189852974
}}</ref> and those by E. Rosinger, Y. Egorov, and R. Robinson.{{citation needed|date=December 2018}}
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 <math>F=F(x)</math> to its smooth 
<math>F_{\rm smooth}</math> and its singular <math>F_{\rm singular}</math> parts. The product of generalized functions <math>F</math> and <math>G</math> appears as

{{NumBlk|:|<math>
FG~=~
F_{\rm smooth}~G_{\rm smooth}~+~
F_{\rm smooth}~G_{\rm singular}~+
F_{\rm singular}~G_{\rm smooth}.</math>|{{EquationRef|1}}}}

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 ({{EquationNote|1}}); in particular, <math>\delta(x)^2=0</math>. 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.<ref name="shirokovAlgebra1dim"/> Few applications of the algebra were suggested.<ref name="goriaga">{{cite journal
|author=O. G. Goryaga
|author2=Yu. M. Shirokov	
|title=Energy levels of an oscillator with singular concentrated potential
|journal=[Theoretical and Mathematical Physics](/source/Theoretical_and_Mathematical_Physics)
|year=1981
|volume=46
|pages=321–324
|doi=10.1007/BF01032729
|issue=3
|bibcode = 1981TMP....46..210G |s2cid=123477107	
}}</ref><ref name="tolok">{{cite journal
|author=G. K. Tolokonnikov
|title=Differential rings used in Shirokov algebras
|journal=[Theoretical and Mathematical Physics](/source/Theoretical_and_Mathematical_Physics)
|volume=53
|issue= 1
|year=1982
|doi=10.1007/BF01014789
|pages=952–954
|bibcode=1982TMP....53..952T
|s2cid=123078052
}}</ref>

===Multiplication of distributions===
The problem of ''multiplication of distributions'', a limitation of the Schwartz distribution theory, becomes serious for [non-linear](/source/non-linear) problems.

Various approaches are used today. The simplest one is based on the definition of generalized function given by Yu. V. Egorov.<ref name="YuVEgorov1990" /> Another approach to construct [associative](/source/associative) [differential algebra](/source/differential_algebra)s is based on  J.-F. Colombeau's construction: see [Colombeau algebra](/source/Colombeau_algebra). These are [factor space](/source/factor_space)s

:<math>G = M / N</math>

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''',
<math>s = \{ a_m:\mathbb N\to\mathbb R, n\mapsto n^m ;~ m\in\mathbb Z \}</math>. Then for any semi normed algebra (E,P), the factor space will be

:<math>G_s(E,P)= \frac{
\{ f\in E^{\mathbb N}\mid\forall p\in P,\exists m\in\mathbb Z:p(f_n)=o(n^m)\}
}{
\{ f\in E^{\mathbb N}\mid\forall p\in P,\forall m\in\mathbb Z:p(f_n)=o(n^m)\}
}.</math>

In particular, for (''E'',&nbsp;''P'')=('''C''',|.|) one gets (Colombeau's) [generalized complex numbers](/source/generalized_number) (which can be "infinitely large" and "infinitesimally small" and still allow for rigorous arithmetics, very similar to [nonstandard number](/source/non-standard_analysis)s). For (''E'',&nbsp;''P'')&nbsp;=&nbsp;(''C<sup>∞</sup>''('''R'''),{''p<sub>k</sub>''}) (where  ''p<sub>k</sub>'' 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](/source/Colombeau_algebra).

===Injection of Schwartz distributions===

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

:''j''(''T'') = (φ<sub>''n''</sub> ∗ ''T'')<sub>''n''</sub>&nbsp;+&nbsp;''N'',

where ∗ is the [convolution](/source/convolution) operation, and

:φ<sub>''n''</sub>(''x'') = ''n'' φ(''nx'').

This injection is ''non-canonical ''in the sense that it depends on the choice of the [mollifier](/source/mollifier) φ, which should be ''C<sup>∞</sup>'', 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'''&nbsp;×&nbsp;''D''('''R'''), with a convenient [filter base](/source/filter_base) on ''D''('''R''') (functions of vanishing [moment](/source/moment_(mathematics))s up to order ''q'').

===Sheaf structure===

If (''E'',''P'') is a (pre-)[sheaf](/source/sheaf_(mathematics)) of semi normed algebras on some topological space ''X'', then ''G<sub>s</sub>''(''E'',&nbsp;''P'') will also have this property. This means that the notion of [restriction](/source/Restriction_(mathematics)) will be defined, which allows to define the [support](/source/support_(mathematics)) of a generalized function w.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 zero).
* For the subsheaf ''E'' (embedded using the canonical (constant) injection), one gets what is called the [singular support](/source/singular_support), i.e., roughly speaking, the closure of the set where the generalized function is not a smooth function (for ''E''&nbsp;=&nbsp;''C''<sup>∞</sup>).

===Microlocal analysis===
{{See also|Microlocal analysis}}
The [Fourier transformation](/source/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](/source/Lars_H%C3%B6rmander)'s ''[wave front set](/source/wave_front_set)'' also for generalized functions.

This has an especially important application in the analysis of [propagation](/source/wave_propagation) of [singularities](/source/Mathematical_singularity).

==Other theories==

These include: the ''convolution quotient'' theory of [Jan Mikusinski](/source/Jan_Mikusinski), based on the [field of fractions](/source/field_of_fractions) of [convolution](/source/convolution) algebras that are [integral domain](/source/integral_domain)s; and the theories of [hyperfunction](/source/hyperfunction)s, based (in their initial conception) on boundary values of [analytic function](/source/analytic_function)s, and now making use of [sheaf theory](/source/sheaf_theory).

==Topological groups==

Bruhat introduced a class of test functions, the [Schwartz–Bruhat function](/source/Schwartz%E2%80%93Bruhat_function)s, on a class of [locally compact group](/source/locally_compact_group)s that goes beyond the [manifold](/source/manifold)s that are the typical [function domain](/source/function_domain)s. The applications are mostly in [number theory](/source/number_theory), particularly to [adelic algebraic group](/source/adelic_algebraic_group)s. [André Weil](/source/Andr%C3%A9_Weil) rewrote [Tate's thesis](/source/Tate's_thesis) in this language, characterizing the [zeta distribution](/source/zeta_distribution_(number_theory)) on the [idele group](/source/idele_group); and has also applied it to the [explicit formula of an L-function](/source/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](/source/vector_bundle). This is on the Schwartz pattern, constructing objects dual to the test objects, smooth sections of a bundle that have [compact support](/source/compact_support). The most developed theory is that of [De Rham current](/source/De_Rham_current)s, dual to [differential form](/source/differential_form)s. These are homological in nature, in the way that differential forms give rise to [De Rham cohomology](/source/De_Rham_cohomology). They can be used to formulate a very general [Stokes' theorem](/source/Stokes'_theorem).

==See also==
* [Beppo-Levi space](/source/Beppo-Levi_space)
* [Dirac delta function](/source/Dirac_delta_function)
* [Generalized eigenfunction](/source/Generalized_eigenfunction)
* [Distribution (mathematics)](/source/Distribution_(mathematics))
* [Hyperfunction](/source/Hyperfunction)
* [Laplacian of the indicator](/source/Laplacian_of_the_indicator)
* [Rigged Hilbert space](/source/Rigged_Hilbert_space)
* [Limit of a distribution](/source/Limit_of_a_distribution)
* [Generalized space](/source/Generalized_space)
* [Ultradistribution](/source/Ultradistribution)

==Books==

*{{cite book |first=L. |last=Schwartz |title=Théorie des distributions |publisher=Hermann |location=Paris |date=1950 |volume=1 |oclc=889264730 }} Vol. 2. {{OCLC|889391733}}
*{{cite book |first=A. |last=Beurling |author-link=Arne Beurling |title=On quasianalyticity and general distributions |type=multigraphed lectures |publisher=Summer Institute, Stanford University  |date=1961 |oclc=679033904 }}
*{{cite book |last1=Gelʹfand |first1=Izrailʹ Moiseevič |last2=Vilenkin |first2=Naum Jakovlevič |author1-link=I.M. Gel'fand |title=Generalized Functions |publisher=Academic Press |volume=I–VI |date=1964 |oclc=728079644 }}
*{{cite book |first=L. |last=Hörmander |title=The Analysis of Linear Partial Differential Operators |publisher=Springer |edition=2nd |orig-date=1990 |isbn=978-3-642-61497-2 |date=2015 |url={{GBurl|aaLrCAAAQBAJ|pg=PR9}}}}
* H. Komatsu, Introduction to the theory of distributions, Second edition, Iwanami Shoten, Tokyo, 1983. <!-- Not found in WorldCat -->
*{{cite book |author-link=Colombeau algebra |first=J.-F. |last=Colombeau |title=New Generalized Functions and Multiplication of Distributions |publisher=Elsevier |date=2000  |orig-date=1983 |isbn=978-0-08-087195-0 |url={{GBurl|7wm-oOMm69EC|pg=PR9}}  }}
*{{cite book |first1=V.S. |last1=Vladimirov |first2=Yu. N. |last2=Drozhzhinov |first3=B.I. |last3=Zav’yalov |title=Tauberian theorems for generalized functions |publisher=Springer |date=2012 |orig-date=1988 |isbn=978-94-009-2831-2 |url={{GBurl|onfvCAAAQBAJ|pg=PR5}} }}
*{{cite book |first=M. |last=Oberguggenberger |title=Multiplication of distributions and applications to partial differential equations |publisher=Longman |date=1992 |isbn=978-0-582-08733-0 |oclc=682138968 }}
*{{cite book |first=M. |last=Morimoto |title=An introduction to Sato's hyperfunctions |publisher=American Mathematical Society |date=1993 |isbn=978-0-8218-8767-7 |url={{GBurl|pcSumZ4aPX0C|pg=PP7}} }}
*{{cite book |first=A.S. |last=Demidov |title=Generalized Functions in Mathematical Physics: Main Ideas and Concepts |publisher=Nova Science |date=2001 |isbn=9781560729051 |url={{GBurl|MFhRr7l3IyAC|p=17}} }}
*{{cite book |first1=M. |last1=Grosser |first2=M. |last2=Kunzinger |first3=Michael |last3=Oberguggenberger |first4=R. |last4=Steinbauer |title=Geometric theory of generalized functions with applications to general relativity |publisher=Springer  |date=2013 |orig-date=2001 |isbn=978-94-015-9845-3 |url={{GBurl|123uCAAAQBAJ|pg=PR5}}}}
*{{cite book |first1=R. |last1=Estrada |first2=R. |last2=Kanwal |title=A distributional approach to asymptotics. Theory and applications |publisher=Birkhäuser Boston |edition=2nd |date=2012 |isbn=978-0-8176-8130-2 |url={{GBurl|X3cECAAAQBAJ|pg=PP7}} }}
*{{cite book |first=V.S. |last=Vladimirov |title=Methods of the theory of generalized functions |publisher=Taylor & Francis |date=2002 |isbn=978-0-415-27356-5 |url={{GBurl|hlumB8fkX0UC|pg=PR5}}}}
*{{cite book |author-link=Hagen Kleinert |first=H. |last=Kleinert |title=Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets |publisher=World Scientific |edition=5th |date=2009 |isbn=9789814273572 |url={{GBurl|VJ1qNz5xYzkC|pg=PR17}}}} ([http://www.physik.fu-berlin.de/~kleinert/b5 online here] {{Webarchive|url=https://web.archive.org/web/20080615134934/http://www.physik.fu-berlin.de/~kleinert/b5 |date=2008-06-15 }}). See Chapter 11 for products of generalized functions.
*{{cite book |first1=S. |last1=Pilipovi |first2=B. |last2=Stankovic |first3=J. |last3=Vindas |title=Asymptotic behavior of generalized functions |publisher=World Scientific |date=2012 |isbn=9789814366847 |url={{GBurl|RidqDQAAQBAJ|pg=PR11}}}}

==References==
<references />
{{Authority control}}

{{DEFAULTSORT:Generalized Function}}
Category:Generalized functions

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