Spectral synthesis for exponentials in weighted $L^2$-spaces

Abstract: We prove that for a some natural class of weights $\K$ and any weighted space $$L^2(w) = \left{ f : (-\pi,\pi) \to \CC \colon\int_{-\pi}^{\pi} {{|f(t)|^2} \over {w(t)}} dt < \infty \right},$$ where $w \in \K$, there exists a complete and minimal system ${e^{i\lambda_nt}}_{n\in \Z}$ of exponentials, which does not admit spectral synthesis.