Asymptotic Properties of Convolution Products of Functions

Edward Omey

Publications de l'Institut Mathématique, (N.S.) 43 (1988) no. 57.

Résumé

The asymptotic behaviour of convolution products of the form $\int_0^x f(x-y)g(y)\,dy$ is studied. From our results we obtain asymptotic expansions of the form $$ R(x) := \int_o^x f(x-y)g(y) dy - f(x)\int^\infty g(y) dy - g(x)\int_0^\infty f(y) dy = O(m(x)). $$ Under rather mild conditions on $f,g$ and $m$ the $O$-term can be calculated more explicitly as $$ R(x)-(f(x-1)-f(x))\int_0^\infty yg(y) dy+(g(x-1) -g(x))\int_0^\infty yf(y) dy + o(m(x)). $$ An application in probability theory is included.