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 Lk, 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 Ln, x denote the differentiation operator
converges for some x¬ 0; then, if f(t) belongs to L(R−1, R) for every R>1,