The neutrix convolution product in \(Z'(m)\) and the exchange formula (Q1273326)

Denoting by \({\mathcal D}'(m)\) the space of distributions defined on the space \({\mathcal D}(m)\) of infinitely differentiable functions, the Fourier transform of a function \(\phi\) in \({\mathcal D}(m)\) is defined by \[ F(\phi)(\sigma)= \psi(\sigma)= \int_{\mathbb{R}^n} \phi(x) e^{i(x,\sigma)} dx, \] where \((x,\sigma)= x_1\sigma_1+\cdots+ x_m\sigma_m\). The Fourier transform \(f\) of a distribution in \({\mathcal D}'(m)\) is called an ultradistribution in \({\mathcal Z}'(m)\), and is a continuous linear functional on \({\mathcal Z}(m)\) defined by Parseval's equation \[ (\widetilde f,\widetilde\phi)= 2\pi(f,\phi),\quad \phi\in{\mathcal D}(m). \] The authors invoke Fourier transform on \({\mathcal D}'(m)\) and an exchange formula to define convolutions of ultradistributions in \({\mathcal Z}'(m)\) in terms of distributions in \({\mathcal D}'(m)\). Some results on convolutions are also established by using the neutrix calculus including a theorem involving the existence of a neutrix convolution \(\widetilde f\otimes\widetilde g\) (\(\widetilde f\), \(\widetilde g\) in \({\mathcal Z}'(m)\)) in \({\mathcal Z}'(m)\).