An exponential basis on a measurable domain of $\Bbb{R}^d$ is a Riesz basis in the form of
$\{ e^{2\pi i \lambda.x} \}_{\lambda\in\Lambda}, $   where $\Lambda$ is a discrete set of $\Bbb{R}^d.$ The problem of proving (or disproving) the existence of such systems on measurable sets is still largely unsolved. For example, the existence of exponential bases on unbounded domains is proved only in very few special cases. Moreover, for most of the domains for which the existence of exponential bases is proved, no explicit expression of such systems is given.

In my talk, I will show explicit examples of exponential bases on finite or infinite unions of intervals. Also, I will describe newly established connections between Vandermonde matrices and exponential bases and prove a stability theorem  for Vandermonde matrices.