# Generalized functions, product of

The product $a f = f a$ of a 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 $a$.

## Contents

### 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). 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 .

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. 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 sets.

How to Cite This Entry:
Generalized functions, product of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Generalized_functions,_product_of&oldid=47072
This article was adapted from an original article by V.S. Vladimirov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article