On the closure of nonclosed systems of functions of Mittag-Leffler type

The closure of a system of functions of Mittag-Leffler type $\{E_\rho(-\lambda_kx;\mu)\}_1^\infty$ ($\operatorname{Re}\lambda_k^\alpha>0$, $1/\rho+1/\alpha=2$) is described in the space $L_{2,\omega}(0,+\infty)$ in the case when the series
$$\sum_{k=1}^\infty\frac{\operatorname{Re}\lambda_k^\alpha}{1+|\lambda_k|^{2\alpha}}$$
is convergent. This result generalizes the well-known theorems of Laurent Schwartz, A. F. Leont'ev and M. M. Dzhrbashyan.
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