# 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 .

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### 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 .

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.