On Hermite Representation of Distributions and Products

The space of tempered distributions $\mathcal{S}^{\prime}$ can be realized as a sequence spaces by means of the Hermite representation theorems. In this paper we introduce and study two new products of tempered distributions based in these Hermite representation. In particular, we obtain the products $[H]\delta=\frac{\delta}{2}$, $[\delta] vp(\frac{1}{x})=-\delta^{\prime}$ and$[\delta^{(r)}]vp(\frac{1}{x})=-\frac{\delta^{(r+1)}}{r+1}$, for $r$ even.