Let \({\mathcal E}_ A(L)\) be the topological vector space of \(C^{\infty}\) functions on \({\mathbb{R}}^ n\) which are approximate solutions to a given convolution equation \(L*f=0\), \(L\in {\mathcal E}'({\mathbb{R}}^ n)\), as in \textit{C. A. Berenstein} and \textit{M. A. Dostal}, ''Analytically uniform spaces and their applications to convolution equations'', (1972; Zbl 0237.47025), and let \(T\subseteq {\mathbb{R}}^ n\) be a linear subvariety. We say that \({\mathcal E}_ A(L)\) is T-quasianalytic if \(\{\) \(f\in {\mathcal E}_ A(L): D^{\alpha}(L^ j*f)=0\) on T for all \(\alpha,j\}=\{0\}\). By employing an extension of Ehrenpreis' Fundamental Principle to convolution equations proved in the above quoted monograph, we show that it is possible to construct \(\mu\in {\mathcal E}'({\mathbb{R}}^{n+1})\), and a weight \(\phi\) such that T-quasianalyticity of \({\mathcal E}_ A(L)\) reduces to the uniqueness of a Cauchy problem for \(\mu\) in the space \({\mathcal E}(\phi)\) of \(C^{\infty}\) functions satisfying certain growth conditions induced by \(\phi\). We also give some explicit conditions on (\(\mu\),\(\phi)\) which make \({\mathcal E}(\phi)\) into a uniqueness space for the Cauchy problem for \(\mu\).