Hereditary completeness for systems of exponentials and reproducing kernels

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• 作者：Anton Baranov^a ; ^{anton.d.baranov@gmail.com} ; Yurii Belov^b ; ^{j_b_juri_belov@mail.ru} ; Alexander Borichev^c ; ^{borichev@cmi.univ-mrs.fr}
• 关键词：Spectral synthesis ; Hereditary completeness ; Paley&ndash ; Wiener space ; de Branges spaces
• 刊名：Advances in Mathematics
• 出版年：2013
• 出版时间：1 March, 2013
• 年：2013
• 卷：235
• 期：Complete
• 页码：525-554
• 全文大小：309 K

文摘

We solve the spectral synthesis problem for exponential systems on an interval. Namely, we prove that any complete and minimal system of exponentials in is hereditarily complete up to a one-dimensional defect. This means that for any partition of the index set, the orthogonal complement to the system , where is the system biorthogonal to , 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.