Generalized functions, product of

The product of a generalized function in and a function is defined by the equation

Here , and for (ordinary) functions in , the product coincides with the ordinary product of the functions and .

Examples.

1) ;

2) .

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:

In order to define the product of two generalized functions and , it is sufficient for them to possess, roughly speaking, the following properties: "non-regularity" of in a neighbourhood of any point must be compensated by corresponding "regularity" of , and conversely; for example, if (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 (one-frequency generalized functions):

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

4) , where is an arbitrary constant. In fact,

But on test functions for which ,

Hence it is natural to put if . Extending this functional to all test functions in , one obtains 4).

5) The definition of the product . The function does not belong to , but it defines regular generalized functions: in , , and in , . They can be consistently extended to generalized functions in , for example, by taking the finite Hadamard part of the divergent integral (renormalizing it)

The generalized function (the renormalized functional for ) depends on the arbitrary parameter . The arbitrariness in the renormalization is the following:

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 ; 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)
[2]	V.S. Vladimirov, "Generalized functions in mathematical physics" , MIR (1979) (Translated from Russian)
[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)

Comments

For another approach to product definitions and generalized functions see [a3].

In general, the product of two generalized functions can be defined unless and for some ( denotes the wave front set of ). See also [a4], Chapt. 8.

References

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