Editing Generalized function (section)

==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]].