Linear Transformations in the Operational Calculus

Abstract

A new development of linear transformations in the one-sided operational calculus is presented. The setting for this is a (noncommutative) ring $\mathcal{R}$ of continuous linear transformations on a familiar test function space from distribution theory. Included in $\mathcal{R}$, both algebraically and topologically, are all the right-sided distributions, all the traditional transformations of the operational calculus, such as the exponential shifts, the dilatations and the algebraic derivative, all translations and multiplications by infinitely differentiable functions, and many other new transformations. The development parallels that given by E. Gesztelyi for linear transformations of the Mikusinski operator field, but is cast in a simpler and more flexible setting. The main tool of the investigation is a representation theorem of the type introduced by V. Dolezal and the results are primarily theorems concerning commutativity properties in $\mathcal{R}$. It is shown that a linear transformation (i) commutes with all translations iff it is a distribution (ii) commutes with differentiation iff it is a distribution (iii) commutes with the algebraic derivative if it is a multiplier and (iv) is a distribution and commutes with every dilatation if it is a number. Because of the latter, it becomes reasonable to define a Laplace transform in $\mathcal{R}$ which encompasses (for right-sided distributions) that givenby D. Price. Some results on inversion in $\mathcal{R}$ are given and a number of unsettled problems, perhaps amenable to solutions in this setting, are mentioned.