International Journal of Mathematics and Mathematical Sciences
Volume 7 (1984), Issue 2, Pages 371-396
doi:10.1155/S0161171284000399
Representation of functions as the Post-Widder inversion operator of generalized functions
1Department of Mathematics, Tribhuwan University, Kirtipur Campus, Kathmandu, Nepal
2Department of Mathematics, University of Central Florida, Orlando 32816, Florida, USA
Received 28 May 1982; Revised 2 January 1984
Copyright © 1984 R. P. Manandhar and L. Debnath. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.