  We solve the spectral synthesis problem for exponential systems on an
interval. Namely, we prove that any complete and minimal system of exponentials
$\{e^{i\lambda_n t}\}$ in $L^2(-a,a)$ is hereditarily complete up to a
one-dimensional defect. This means that there is at most one (up to a constant
factor) function $f$ which is orthogonal to all the summands in its formal
Fourier series $\sum_n (f,\tilde e_n) e^{i\lambda_n t}$, where $\{\tilde e_n\}$
is the system biorthogonal to $\{e^{i\lambda_n t}\}$. However, this
one-dimensional defect is possible and, thus, there exist nonhereditarily
complete exponential systems. Analogous results are obtained for systems of
reproducing kernels in de Branges spaces. For a wide class of de Branges spaces
we construct nonhereditarily complete systems of reproducing kernels, thus
answering a question posed by N. Nikolski.
