Spectral synthesis for exponential systems

We consider the problem whether it is possible to recover an arbitrary function in $L^2$ from its generalized Fourier series with respect to some complete and minimal system of exponentials (it is known as the spectral synthesis problem). The main result is as follows:
1. There exist complete and minimal systems of exponentials for which the spectral synthesis is not possible.
2. The spectral synthesis for exponential systems always holds up to a one-dimensional defect.
We plan to discuss the idea of the proof of these results.
The talk is based on a joint work with Yurii Belov and Alexander Borichev.