The Division of Distributions by the Independent Variable

Abstract

I n the theory of distributions the multiplication of a distribution by a holomorphic function without, singularities is well defined. For instance, we havewhere 6 is DIRAC'S delta distribution. Sometimes, more general products are defined as cf.[l], p. 249). But if one tries to calculate with l/t in a naive way, one gets nequalit,ies like so that there is no associativity.The main result of this paper is the following: It may be arranged, that one has quite well associativity, but no commutativity. All considerat.ions are essentially based on the theory of right invertible operators in the framework of the general operational calculus (cf. {2], [7]). To put things clear, we restrict ourselves at the beginning to a simple subclass of distributions. An isomorphism correslmnding to the LAPLACE transform shows, that our definitions are quite natural. It is easily possible to extend the results to wider classes of distributions by using a H -4 3 1 .~~ basis, converging series with a suitable topology (cf. [9]) or formal series (cf. 161). Such extensions are sketched at the end of the paper. Preliminaries. Let 23 be the least vector space generated by one element b i o and by two linear right invertible operators D, T with (1) DT -TD = 1, TDb = Oin such a way. that D and T map 23 onto 23. The field R of the coefficients of % shall have the characteristic zero. So we can assume, that the field of rational numbers is embeded in R, and that 1 is the unit operator.

A Two-Sided Operational Calculus

“…The definition of this integral for the remaining elements of B Q can be done by means of a Hamel basis (cf. [3]). If we also consider translations <f>(p+/l) of the elements $(p)€B" with arbitra--At ry complex numbers A, and multiplications by e~ in A Q (remark that there is no one-to-one correspondence between them), we get an operational calculus of almost the same generality as in the book of Amerbaev Ql].…”