# Generalized functions, product of

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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)