Convolution over the Spaces S′k

In this paper, we prove that for f ∈ S′k, and g ∈ S′k, k ∈ Z, k < 0, the extension from D to Sk of the convolution f ∗ g, defined according to the lines of Schwartz, Horváth, and Dierolf and Voigt, agrees with the functional f#g ∈ S′k, given by 〈f(x), 〈g(y), φ(x + y)〉〉, φ ∈ Sk, where S′k is the dual of the space Sk introduced by Horváth.