Proposition 1

(i) If $t \in T_g^{(s)}(G)$ and $t' \in T_{g'}^{(s')}(G)$, then $t * t' \in T_{gg'}^{(s+s')}(G)$.

(ii) If $t$ or $t$ has no constant term, $t * t'$ has no constant term.

(iii) $\varepsilon_g * \varepsilon_{g'} = \varepsilon_{gg'}$.

(iv) Let $t \in T_g^{(s)}(G)$, $t' \in T_{g'}^{(s')}(G)$ and let $f$ be a function of class $C^{s+s'}$ in an open neighbourhood of $gg'$ with values in a Hausdorff polynormed space. Then

This follows from Differentiable and Analytic Manifolds, R, 13.4.1, 13.2.3 and 13.4.4.

Suppose that K = \mathbf{R} or \mathbf{C} and that G is finite-dimensional. Then G is locally compact. If t, t' are point measures, the definition of t \* t' agrees with that of Integration, Chapter VIII, § 1. We shall see later that the convolution product of measures and that of point distributions are two special cases of the convolution product of distributions which are not necessarily point distributions.

Let \mathscr{T}^{(\infty)}(G) be the direct sum of the T_g^{(\infty)}(G) for g \in G (cf. Differentiable and Analytic Manifolds, R, 13.6.1). We define the convolution product in \mathscr{T}^{(\infty)}(G) as the bilinear mapping of \mathscr{T}^{(\infty)}(G) \times \mathscr{T}^{(\infty)}(G) into \mathscr{T}^{(\infty)}(G) extending the convolution product of Definition 1. We also denote it by . Thus \mathscr{T}^{(\infty)}(G) has an algebra structure filtered by the \mathscr{T}^{(s)}(G). The subalgebra \mathscr{T}^{(0)}(G) = \bigoplus{g \in G} T_g^{(0)}(G) is identified with the algebra K^{(G)} of the group G over K.