  The complex exponentials with integer frequencies form a basis for the space
of square integrable functions on the unit interval. We analyze whether the
basis property is maintained if the support of the complex exponentials is
restricted to possibly overlapping subsets of the unit interval. We show, for
example, that if $S_1, \ldots, S_K \subset [0,1]$ are finite unions of
intervals with rational endpoints that cover the unit interval, then there
exists a partition of $\mathbb{Z}$ into sets $\Lambda_1, \ldots, \Lambda_K$
such that $\bigcup_{k=1}^K \{ e^{2\pi i \lambda (\cdot)} \chi_{S_k} : \lambda
\in \Lambda_k \}$ is a Riesz basis for $L^2[0,1]$. Here, $\chi_S$ denotes the
characteristic function of $S$.
