Approximation of probability distributions by convex mixtures of Gaussian measures

Let $\mathcal{A_{+}}=\{a=(a_{n})\in\bigcap_{p>1}\ell_{p}:a_{n}>0, \forall n\in\mathbb{N}\}$ and let $\{\phi_{j}\}_{j=1}^{\infty}$ be an enumeration of all normal distributions with mean a rational number and variance $\frac{1}{n^{2}}, n=1,2\dots$. We prove that there exists an $a\in\mathcal{A_{+}}$ such that every probability density function, continuous, with compact support in $\mathbb{R}$, can be approximated in $L^{1}$ and $L^{\infty}$ norm simultaneously by the averages $\frac{1}{\sum_{j=1}^{n}a_{j}} \sum_{j=1}^{n}a_{j}\phi_{j}$. The set of such sequences is a dense $G_{\delta}$ set in $\mathcal{A_{+}}$ and contains a dense positive cone.