Hereditary completeness for systems of exponentials and reproducing kernels

We solve the spectral synthesis problem for exponential systems on an interval. Namely, we prove that any complete and minimal system of exponentials {eiλnt}n∈N{eiλnt}n∈N in L2(−a,a)L2(−a,a) is hereditarily complete up to a one-dimensional defect. This means that for any partition N=N1∪N2N=N1∪N2 of the index set, the orthogonal complement to the system {eiλnt}n∈N1∪{en′}n∈N2, where {en′} is the system biorthogonal to {eiλnt}{eiλnt}, is at most one-dimensional. 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.