Polyconvolution of Hartley integral transforms $H_2$ and integral equations

Abstract

We construct and study a new polyconvolution $\ast \left(f,g,h\right)\left(x\right)$ of functions $f$, $g$ and $h$ for the Hartley integral transform $H_2$. We will show that polyconvolution satisfies the following factorization equality:

$$H_2\left(f\underset{2}{\ast }g\right)\left(y\right)=\left(H_{2}f\right)\left(y\right)\left(H_{2}g\right)\left(y\right)\left(H_{2}h\right)\left(y\right),\forall y\in ℝ.$$

We prove the existence of this polyconvolution in the space $L\left(ℝ\right)$. As examples, applications to solve an integral equation of polyconvolution type and a system of integral equations of polyconvolution type are presented.