Yurii Shirokov

“ For this, we note that the product (27) (in contrast to the bilinear form (23) ) is also defined for functions that do not decrease at infinity and, in particular, for . At the same time,
. It is therefore convenient to denote
and assume that the functions and behave in accordance with (3) and (4) in functionals of type and in accordance with (9) , (10) , and (30) in functionals of the type . In such a way, (29) becomes an identity.
The following simplification is suggested by the fact that for any infinitely smooth functions in and in , we have
where is the ordinary delta function. ”

“ Regilarizing functionals Functions allow to introduce new functionals on functions of the space (which here play the part of test functions) . The functionals of a new type have support concentrated at the same point but they differ from . We denote these functlonals by and ; we define them as follows:
, where
We emphasize that the limits in (6) and (7) are made after the limits in (3) and (4) ; it is the trick which allows to make the algebra assotiative. For example,
where is the Kronecker diagonal tensor. ”

“ Both spaces consist of complex functions of a single variable that are finite, with all their derivatives, everywhere except at the point , and at have finite limits from the left and the right for the functions themselves and for all derivatives.
Slow rise and fast decay At infinity, functions in , together with all their derivatives, can have slow (i.e., not faster than polynomial) growth, while functions in decrease rapidly with all their derivatives (i.e., faster than any polynomial) . ”

“ The simplest variant of this model is one-dimensional motion of a nonrelativistic particle in the field of such a potential. It is true that for a one-dimensional potential proportional to , the model is superrenormalizable, but for potentials contacting , , etc., one can obtain various nonsuperrenormalizable models, Including nonrenormallzable models. ”