Convolution over the spaces \({\mathcal S}'_k\) (Q1892247)

The authors prove that for \(f\in {\mathcal S}_k'\) and \(g\in {\mathcal S}_k'\), \(k\in \mathbb{Z}\), \(k<0\), the extension from \({\mathcal D}\) to \({\mathcal S}_k\) of the convolution \(f* g\), defined according to the lines of Schwartz, Horváth, and Dierolf and Voigt, agrees with the functional \(f\# g\in {\mathcal S}_k'\), given by \(\langle f(x), \langle g(y), \varphi(x+ y)\rangle \rangle\), \(\varphi\in {\mathcal S}_k\), where \({\mathcal S}_k'\) is the dual of the space \({\mathcal S}_k\) introduced by Horváth.