Representation of functions as the Post-Widder inversion operator of generalized functions

Abstract

A study is made of the Post-Widder inversion operator to a class of generalized functions in the sense of distributional convergence. Necessary and sufficient conditions are proved for a given function to have the representation as the rth operate of the Post-Widder inversion operator of generalized functions. Some representation theorems are also proved. Certain results concerning the testing function space and its dual are established. A fundamental theorem regarding the existence of the real inversion operator (1.6) with r=0 is proved in section 4. A classical inversion theory for the Post-Widder inversion operator with a few other theorems which are fundamental to the representation theory is also developed in this paper.

Keywords

• Post-Widder inversion operator
• representation theorems
• testing function space and its dual
• generalized functions.