Generalized Gâteaux and Fréchet derivatives in convolution algebras

Abstract

Let be a commutative normed algebra whose elements are functions on a Banach space, and in which the product is of convolution type. We show how , under reasonable assumptions upon its semi-group of translation operators and rather severe restrictions upon its semi-simplicity, can be renormed so that its completion contains mean Gâteaux or Fréchet differentials of all its members. In this way the function algebra can be embedded in an algebra of generalized functions defined as strong limits of functions, which admits strong differentiation as an everywhere-defined operation. The differentials show most of the properties of point-wise Gâteaux differentials. The completion of is an example of what I have called in (3) an ‘inflation’.