Exponential approximation and meromorphic interpolation

We establish a relation between the approximation in $L^2[-\pi ,\pi ]$ by exponentials with the set of frequencies of Beurling–Malliavin density less than $1$ and the meromorphic interpolation at $\mathbb{Z}$. Furthermore, we show that typical $L^2[-\pi ,\pi ]$ functions admit such an approximation.