# 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 |

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Generalized functions, product of.

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