Hereditary completeness for systems of exponentials and reproducing kernels

By A. Baranov, Y. Belov, A. Borichev. 2011

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Abstract 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 λ n t } n ∈ N in L 2 ( − a , a ) is hereditarily complete up to a one-dimensional defect. This means that for any partition N = N 1 ∪ N 2 of the index set, the orthogonal complement to the system { e i λ n t } n ∈ N 1 ∪ { e n ′ } n ∈ N 2 , where { e n ′ } is the system biorthogonal to { e i λ n t } , 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.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f4cb878c3617be49d34f1c4c21cb129dedc3ed31