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The product  $  a f = f a $
of a [[Generalized function|generalized function]]  $  f $
in  $  D  ^  \prime  ( O) $
and a function  $  a \in C  ^  \infty  ( O) $
is defined by the equation

$$ 
( a f , \phi )  =  ( f , a \phi ) ,\ \ 
\phi \in D ( O) .
$$

Here  $  a f \in D  ^  \prime  ( O) $, 
and for (ordinary) functions  $  f $
in  $  L _ { \mathop{\rm loc}  }  ^ {1} ( O) $, 
the product  $  a f $
coincides with the ordinary product of the functions  $  f $
and  $  a $.

===Examples.===

1)  $  a ( x) \delta ( x) = a ( 0) \delta ( x) $;

2)  $  x {\mathcal P}  ( 1 / x ) = 1 $.

However, this product operation cannot be extended to arbitrary generalized functions in such a way that it is associative and commutative, otherwise there would be the contradiction:

$$ 
( x \delta ( x) ) {\mathcal P} \left ( 
\frac{1}{x}
 \right )  = \ 
0 {\mathcal P} \left ( 
\frac{1}{x}
 \right )  =  0 ,
$$

$$ 
( x \delta ( x) ) {\mathcal P} \left ( 
\frac{1}{x}
 \right )  = \ 
( \delta ( x) x ) {\mathcal P} \left ( 
\frac{1}{x}
 \right )  =  \delta ( x)
\left ( x {\mathcal P} \left ( 
\frac{1}{x}
 \right )  \right )  =  \delta ( x) .
$$

In order to define the product of two generalized functions  $  f $
and  $  g $, 
it is sufficient for them to possess, roughly speaking, the following properties:  "non-regularity"  of  $  f $
in a neighbourhood of any point must be compensated by corresponding  "regularity"  of  $  g $, 
and conversely; for example, if  $  \textrm{ sing  supp  }  f \cap \textrm{ sing  supp  }  g = \emptyset $(
see [[Support of a generalized function|Support of a generalized function]]). A product can be defined in certain classes of generalized functions, but it may turn out not to be uniquely determined.

===Examples.===

3) The boundary values of the algebra of holomorphic functions  $  H ( C) $(
one-frequency generalized functions):

$$ 
f ( x + i 0 ) g ( x + i 0 )  = \ 
\lim\limits _
{\begin{array}{c}
y \rightarrow 0, \\
 y \in C 
\end{array}
 } \ 
f ( x + i y ) g ( x + i y ) \ \ 
 \mathop{\rm in}  S  ^  \prime  .
$$

They form an associative and commutative algebra with an identity [[#References|[2]]].

4)  $  \delta  ^ {2} ( x) = c \delta ( x) $, 
where  $  c $
is an arbitrary constant. In fact,

$$ 
\delta _  \epsilon  ( x)  = \ 

\frac \epsilon {\pi ( x  ^ {2} + \epsilon  ^ {2} ) }
  \rightarrow  \delta ( x) ,\ \ 
\epsilon \downarrow 0 ,\   \mathop{\rm in}  D  ^  \prime  .
$$

But on test functions  $  \phi $
for which  $  \phi ( 0) = 0 $,

$$ 
( \delta _  \epsilon   ^ {2} , \phi )  = \ 
\int\limits _ {- \infty } ^ { {+ }  \infty }

\frac{\epsilon  ^ {2} }{\pi  ^ {2} ( x  ^ {2} + \epsilon  ^ {2} )  ^ {2} }

\phi ( x)  d x \rightarrow 0 ,\ \ 
\epsilon \downarrow 0 .
$$

Hence it is natural to put  $  ( \delta  ^ {2} , \phi ) = 0 $
if  $  \phi \in D, $
$  \phi ( 0) = 0 $. 
Extending this functional to all test functions  $  \phi $
in  $  D $, 
one obtains 4).

5) The definition of the product  $  \theta ( x) / x $. 
The function  $  \theta ( x) / x $
does not belong to  $  L _ { \mathop{\rm loc}  }  ^ {1} ( \mathbf R  ^ {1)} $, 
but it defines regular generalized functions:  $  0 $
in  $  D  ^  \prime  $, 
$  x < 0 $, 
and  $  1 / x $
in  $  D  ^  \prime  $, 
$  x > 0 $. 
They can be consistently extended to generalized functions in  $  D  ^  \prime  $, 
for example, by taking the finite Hadamard part of the divergent integral (renormalizing it)

$$ 
\left ( \left (

\frac{\theta ( x) }{x}

\right ) _ {M} , \phi
\right )  = \ 
\int\limits _ { 0 } ^ { M } 

\frac{\phi ( x) - \phi ( 0) }{x}
  d x +
\int\limits _ { M } ^  \infty  

\frac{\phi ( x) }{x}
  dx ,\ \ 
\phi \in D .
$$

The generalized function  $  ( \theta ( x) / x ) _ {M} $(
the renormalized functional for  $  \theta ( x) / x $) 
depends on the arbitrary parameter  $  M > 0 $. 
The arbitrariness in the renormalization is the following:

$$ 
\left (

\frac{\theta ( x) }{x}

\right ) _ {M} -
\left (

\frac{\theta ( x) }{x}

\right ) _ {M _ {1}  }  = \ 
 \mathop{\rm ln}  
\frac{M _ {1} }{M}

\delta ( x) .
$$

These ideas lead to the procedure of renormalization of Feynman amplitudes in [[Quantum field theory|quantum field theory]]. The renormalization constants (for examples, masses and charges) appear as arbitrary constants, like  $   \mathop{\rm ln} ( M _ {1} / M ) $; 
the most general definition of a product of generalized functions is given in terms of [[Wave front|wave front]] sets.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L. Schwartz,   "Théorie des distributions" , '''1–2''' , Hermann  (1950–1951)  {{MR|2067351}} {{MR|0209834}} {{MR|0117544}} {{MR|0107812}} {{MR|0041345}} {{MR|0035918}} {{MR|0032815}} {{MR|0031106}} {{MR|0025615}} {{ZBL|0962.46025}} {{ZBL|0653.46037}} {{ZBL|0399.46028}} {{ZBL|0149.09501}} {{ZBL|0085.09703}} {{ZBL|0089.09801}} {{ZBL|0089.09601}} {{ZBL|0078.11003}} {{ZBL|0042.11405}} {{ZBL|0037.07301}} {{ZBL|0039.33201}} {{ZBL|0030.12601}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.S. Vladimirov,   "Generalized functions in mathematical physics" , MIR  (1979)  (Translated from Russian)  {{MR|0564116}} {{MR|0549767}} {{ZBL|0515.46034}} {{ZBL|0515.46033}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N.N. Bogolyubov,   O.S. Parasyuk,   "Ueber die Multiplication der Kausalfunktionen in der Quantentheorie der Felder"  ''Acta Math.'' , '''97'''  (1957)  pp. 227–266  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  C. Hepp,   "Théorie de la renormalisation" , ''Lect. notes in physics'' , '''2''' , Springer  (1969)  {{MR|0277208}} {{ZBL|}} </TD></TR></table>

====Comments====
For another approach to product definitions and generalized functions see [[#References|[a3]]].

In general, the product of two generalized functions  $  f , g \in D  ^  \prime  ( 0) $
can be defined unless  $  ( x , \xi ) \in  \mathop{\rm WF} ( f  ) $
and  $  ( x , - \xi ) \in  \mathop{\rm WF} ( g) $
for some  $  ( x , \xi ) $(
$   \mathop{\rm WF} ( f  ) $
denotes the [[Wave front|wave front]] set of  $  f  $). 
See also [[#References|[a4]]], Chapt. 8.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.F. Colombeau,   "New generalized functions and multiplication of distributions" , North-Holland  (1984)  {{MR|0738781}} {{ZBL|0532.46019}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  K. Keller,   "Analytic regularizations, finite part prescriptions and products of distributions"  ''Math. Ann.'' , '''236'''  (1978)  pp. 49–84  {{MR|0499170}} {{ZBL|0386.46036}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  T.H. Koornwinder,   J.J. Looder,   "Generalized functions"  P.L. Butzer (ed.)  R.L. Stens (ed.)  B. Sz.-Nagy (ed.) , ''An aniversary volume on approximation theory and functional analysis'' , Birkhäuser  (1984)  pp. 151–164  {{MR|0820519}} {{ZBL|0556.46021}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  L.V. Hörmander,   "The analysis of linear partial differential operators" , '''1''' , Springer  (1983)  pp. §7.7  {{MR|0717035}} {{MR|0705278}} {{ZBL|0521.35002}} {{ZBL|0521.35001}} </TD></TR></table>