On absolute completeness of systems of exponentials on a closed interval

Let $\Lambda=\{\lambda_i\}$ be a sequence of points in the complex plane, and $M=\{m_i\}$ a sequence of positive numbers. Problem: under what relations between $\Lambda$ and $M$ can any function in $C[a,b]$ be approximated in the uniform norm by finite linear combinations $\sum a_ie^{\lambda_ix}$ of exponentials with the coefficient restriction $|a_i|\leqslant C_fm_i$. Here $C_f$ depends only on $f$.
An exact solution of the problem is given under the assumption that $\big|\frac{\operatorname{Im}\lambda_i}{\operatorname{Re}\lambda_i}\big|\leqslant\text{Const}$.
Bibliography: 26 titles.