Tangential boundary values of Laplace transforms. Applications to Muntz–Szasz type approximation

We consider the Laplace transforms (LT) of functions in $L^q(\mathbb R_+)$, $1<q\leqslant 2$, with a slowly varying weight. We prove that if the weight satisfies certain conditions, then each LT of this class has tangential boundary values almost everywhere on the imaginary axis, and the structure of the corresponding neighbourhoods depends on the weight only. This result is applied to distinguish a wide class of weighted $L^p$ spaces on the half-line such that the Szasz condition is not necessary for the completeness of the system $\exp(-\lambda_n t)$ in these spaces.