Hereditary completeness for systems of exponentials and reproducing kernels

Details Hereditary completeness for systems of exponentials and reproducing kernels Hereditary completeness for systems of exponentials and reproducing kernels

2011-12-23

arxiv.org/abs/1112.5551v2

Mathematics

Complex Variables

35 pages. Major changes in Sections 4 and 5. An example of a nonhereditarily complete system of exponentials is constructed

Scientific paper

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.

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