Tauberian-type theorems with application to the Stieltjes transformation (Q2469024)

Summary: In the first part, we define the space \(L'(r)\) and the modified Stieltjes transformation introduced by \textit{J.\,Lavoine} and \textit{O.\,P.\thinspace Misra} [Math.\ Proc.\ Camb.\ Philos.\ Soc.\ 86, 287--293 (1979; Zbl 0414.46028)] and [\textit{O.\,I.\thinspace Marichev}, ``Handbook of Integral Transforms of Higher Transcendental Functions: Theory and Algorithmic Tables'' (Ellis Horwood Series in Mathematics and its Applications; Chichester: Ellis Horwood \& New York: Halsted Press) (1983; Zbl 0494.33001)], respectively. In the second part of the paper, we extend Tauberian-type theorems for the distributional Stieltjes transformations to the distributional modified Stieltjes transformations.