A Class of Dilation Integral Equations

Abstract

We present a class of dilation integral equations. The equations in this class depend on a dilation parameter $a\in\mathbb{R}.$ The existence of non trivial solutions in $L^1(\mathbb{R})$ is studied as a function of the dilation parameter. The main result establishes the non existence of these solutions for $|a|<1,$ a necessary and sufficient condition for the existence of solutions with non vanishing integrals in case $|a|>1,$ and sufficient conditions for these equations to have no solutions but the trivial one or to have an infinitude of non trivial solutions in case $|a|=1.$ In all these cases, the dimension of the space of $L^1(\mathbb{R})$-solutions is determined. When $|a|>1$ we have succeeded in writing the frequency domain representation of the solutions as convergent infinite products.