On multiplication of distributions generated by the Pommiez operator (Q2314131)

Given an open subset \(\Omega \subseteq \mathbb{R}\), the product of a distribution \(u \in \mathscr{D}'(\Omega)\) and an infinitely differentiable function \(v \in C^\infty(\Omega)\) can be represented via the Fourier-Laplace transform \(\mathscr{F} := \mathscr{D}(\Omega) \to \mathscr{A}_\Omega\), as \(\mathscr{F}'((\mathscr{F}')^{-1}(u) \mathbin{\widetilde \otimes} (\mathscr{F}')^{-1}(v))\) where \(\widetilde \otimes\) is the convolution \((\varphi \mathbin{\widetilde \otimes} \psi)(f) := \psi_z ( \varphi_t(f(t+z))\), defined for elements \(\varphi, \psi \in \mathscr{A}_\Omega'\) such that \(\mathscr{F}'(\varphi)\) is infinitely differentiable, and \(\mathscr{A}_\Omega\) is the usual inductive limit of weighted spaces of entire functions. In the present paper, the authors replace the translation operator in the convolution by an operator \(T_z\) such that the product extends to a commutative associative multiplication on the whole space \(\mathscr{D}'(\Omega)\); this operator is given by the translation operator for the Pommiez operator, defined by \(T_z(f)(t) = ( t f(t) g_0(z) - z f(z) g_0(t)) / (t-z)\) for \(t \ne z\) and \(z g_0(z) f'(z) - z f(z) g_0'(z) + f(z) g_0(z)\) for \(t=z\), where \(g_0\) is the Fourier-Laplace transform of a test function having integral one. The authors obtain an analytic realization of this multiplication and give some examples.