A convolution integral equation solved by Laplace transformations

摘要

We consider the integral equation p(t) = ∫0tK(t−τ)p(τ)dτ+r(t) where both K(t) and r(t) behave as exp(αt3) as t → ∞ (α > 0). So straightforward application of the Laplace transform technique is not possible. By introducing a complex parameter the equation is solved in the complex domain. Analytic continuation with respect to this parameter yields the desired solution. For a particular example (which arose in a statistical problem on estimating monotone densities) we describe the construction of the explicit solution of the equation.