Editing Generalized function (section)

===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>