On the Stieltjes transform of generalized functions

Abstract

If f ( t ) belongs to L (0, R ) for every positive R and is such that the integral converges for x > 0, then F ( s ) exists for complex s ( s ╪ 0) not lying on the negative real axis and for any positive ξ at which f (ξ+) and f (ξ−) both exist. We define an operator L k, t [ F ( x )]by Under the above conditions on f (t), it is known that for all points t of the Lebesgue set for the function f (t), Let L n, x denote the differentiation operator Suppose that converges for some x¬ 0; then, if f ( t ) belongs to L ( R −1 , R ) for every R >1,