Editing Generalized function

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

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

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 were questionable 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.

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 each 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]].

During the late 1920s and 1930s further basic steps were taken. The [[Dirac delta function]] was boldly defined by [[Paul Dirac]] (an aspect of his [[scientific formalism]]); this was to treat [[measure (mathematics)|measures]], thought of as densities (such as [[charge density]]) like genuine functions. [[Sergei Sobolev]], working in [[partial differential equation theory]], defined the first rigorous theory of generalized functions in order to define [[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. |url=https://www.worldcat.org/oclc/44675353 |title=Elements of the theory of functions and functional analysis |last2=Fomin |first2=S. V. |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]] and [[Kurt Friedrichs]]. Sobolev's work was extended by [[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 [[distribution (mathematics)|distributions]] developed by [[Laurent Schwartz]], systematically working out the principle of [[dual space|duality]] for [[topological vector space]]s. Its main rival in [[applied mathematics]] is [[mollifier]] theory, which uses sequences of smooth approximations (the '[[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]]s, they do not form an [[algebra]]. For example, it is meaningless to square the [[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]] 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 [[Hagen Kleinert|H. 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}}</ref> The result is equivalent to what can be derived from
[[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]]
|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]]
|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]]
|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]] 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]] [[differential algebra]]s is based on  J.-F. Colombeau's construction: see [[Colombeau algebra]]. These are [[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 number|generalized complex numbers]] (which can be "infinitely large" and "infinitesimally small" and still allow for rigorous arithmetics, very similar to [[non-standard analysis|nonstandard number]]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 algebra|Colombeau's simplified 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]] 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]] 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]] on ''D''('''R''') (functions of vanishing [[moment (mathematics)|moment]]s up to order ''q'').

===Sheaf structure===

If (''E'',''P'') is a (pre-)[[sheaf (mathematics)|sheaf]] 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 (mathematics)|restriction]] will be defined, which allows to define the [[support (mathematics)|support]] 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]], 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]] 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 has an especially important application in the analysis of [[wave propagation|propagation]] of [[Mathematical singularity|singularities]].

==Other theories==

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

==Topological groups==

Bruhat introduced a class of [[test function]]s, the [[Schwartz–Bruhat function]]s, on a class of [[locally compact group]]s that goes beyond the [[manifold]]s that are the typical [[function domain]]s. The applications are mostly in [[number theory]], particularly to [[adelic algebraic group]]s. [[André Weil]] rewrote [[Tate's thesis]] in this language, characterizing the [[zeta distribution (number theory)|zeta distribution]] on the [[idele group]]; 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]]. 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 current]]s, dual to [[differential form]]s. 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]].

==See also==
* [[Beppo-Levi space]]
* [[Dirac delta function]]
* [[Generalized eigenfunction]]
* [[Distribution (mathematics)]]
* [[Hyperfunction]]
* [[Laplacian of the indicator]]
* [[Rigged Hilbert space]]
* [[Limit of a distribution]]
* [[Generalized space]]
* [[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 |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]). 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| ]]