International Journal of Mathematics and Mathematical Sciences
Volume 10 (1987), Issue 4, Pages 641-670
Recent developments on the Stieltjes transform of generalized functions
1Department of Mathematics, Banaras Hindu University, Varanasi 5, India
2Department of Mathematics, University of Central Florida, Orlando 32816, FL, USA
Received 11 November 1985; Revised 20 December 1986
Copyright © 1987 Ram Sankar Pathak and Lokenath 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.
This paper is concerned with recent developments on the Stieltjes transform
of generalized functions. Sections 1 and 2 give a very brief introduction to the
subject and the Stieltjes transform of ordinary functions with an emphasis to the
inversion theorems. The Stieltjes transform of generalized functions is described
in section 3 with a special attention to the inversion theorems of this transform.
Sections 4 and 5 deal with the adjoint and kernel methods used for the development of
the Stieltjes transform of generalized functions. The real and complex inversion
theorems are discussed in sections 6 and 7. The Poisson transform of generalized
functions, the iteration of the Laplace transform and the iterated Stieltjes transfrom
are included in sections 8, 9 and 10. The Stieltjes transforms of different
orders and the fractional order integration and further generalizations of the
Stieltjes transform are discussed in sections 11 and 12. Sections 13, 14 and 15 are
devoted to Abelian theorems, initial-value and final-value results. Some applications
of the Stieltjes transforms are discussed in section 16. The final section deals with
some open questions and unsolved problems. Many important and recent references are
listed at the end.