Hereditary completeness for systems of exponentials in weighted $L^2$-spaces

We prove that for a weight $w$, which has at least polynomial decay, there exists a complete and minimal system $\{e^{i{\lambda }_{n}t}{\}}_{n\in \mathbb{N}}$ of exponentials in weighted space $L^2(w)$ on $(-\pi ,\pi )$, which is not hereditarily complete.