On a generalization of W.A.J. Luxemburg's asymptotic problem concerning the Laplace transform

In this paper we prove a more general case of Luxemburg's asymptotic problem concerning the Laplace transform: The problem deals with the conservation of a certain asymptotic behavior of a function at infinity, under analytic transformation of its Laplace transform. The theory of commutative Banach algebras tells us that the problem is equivalent to a family of special cases of the original problem, viz. a set of convolution integral equations, parametrized by a complex variable λ. For ∥ λ ∥ large enough, we may use Luxemburg's original result, and for other λ we modify the integral equations, and apply a modification of Luxemburg's result.