A Mollification Framework for Improperly Posed Problems

1. Introduction Various authors have explored the role of molliication in the solution of improperly posed problems such as numerical diierentiation and the numerical solution of rst kind integral equations. For example, Murio 13], exploiting earlier suggestions of Manselli and Miller 11] as well as Vasin 16], has shown how molliied numerical diierentiators can be constructed for the diierentiation of observational data. But, the underlying regularity requires the existence of the second derivative of the function being diierentiated. On the other hand, HH ao et al. 4] proposed and examined a molliication procedure where the regularity reduced to only requiring the existence of the fractional derivative being evaluated. Independently, Anderssen and de Hoog 2] examined the use of multi-point (moving average) nite diierence formulas for the numerical diierentiation of observational data, and show how the length of such formulas must be related to the size of the discretization of the data before convergence and stability (bounded ampliication error) can be guaranteed as the size of the discretization is decreased. This represents an explicit exempliication of the situation which occurs in molliication where some appropriate measure of the extent of the molliication must be related in some suitable way to the discretization of the data before convergence and stability results can be derived. However, to-date, such investigations have not explicitly exploited the natural translation and dilation properties of fractional diierentiation operators. The purpose of this paper is to show how, using the classical semi-group theory of Hille and Phillips 8], a quite general theory can be constructed for the molliication of operator equations