Error Bounds for Asymptotic Expansions of Laplace Convolutions

Asymptotic expansions are derived for the Laplace convolution $(f * g)(x)$ as $x \to \infty $, where f and g have asymptotic power series representation in descending powers of t. Bounds are also constructed for the error terms associated with these expansions. Similar results are given for the convolution integrals \[ \int_0^\infty {f(t)g(x + t)dt} \qquad {\text{and}}\qquad \int_0^\infty {f(t)g(x - t)dt} \] as $x \to \infty $. These results can be used in the study of asymptotic solutions to the renewal equation and the Wiener–Hopf equations.