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 $.

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 [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. 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.

References

[1]	L. Schwartz, "Théorie des distributions" , 1–2 , Hermann (1950–1951) MR2067351 MR0209834 MR0117544 MR0107812 MR0041345 MR0035918 MR0032815 MR0031106 MR0025615 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
[2]	V.S. Vladimirov, "Generalized functions in mathematical physics" , MIR (1979) (Translated from Russian) MR0564116 MR0549767 Zbl 0515.46034 Zbl 0515.46033
[3]	N.N. Bogolyubov, O.S. Parasyuk, "Ueber die Multiplication der Kausalfunktionen in der Quantentheorie der Felder" Acta Math. , 97 (1957) pp. 227–266
[4]	C. Hepp, "Théorie de la renormalisation" , Lect. notes in physics , 2 , Springer (1969) MR0277208

Comments

For another approach to product definitions and generalized functions see [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 set of $ f $). See also [a4], Chapt. 8.

References

[a1]	J.F. Colombeau, "New generalized functions and multiplication of distributions" , North-Holland (1984) MR0738781 Zbl 0532.46019
[a2]	K. Keller, "Analytic regularizations, finite part prescriptions and products of distributions" Math. Ann. , 236 (1978) pp. 49–84 MR0499170 Zbl 0386.46036
[a3]	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 MR0820519 Zbl 0556.46021
[a4]	L.V. Hörmander, "The analysis of linear partial differential operators" , 1 , Springer (1983) pp. §7.7 MR0717035 MR0705278 Zbl 0521.35002 Zbl 0521.35001