Spectral synthesis in de Branges spaces

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• 作者：Anton Baranov (1) (2)
  Yurii Belov (3)
  Alexander Borichev (4)
  
  1. Department of Mathematics and Mechanics ; St. Petersburg State University ; St. Petersburg ; Russia
  2. National Research University Higher School of Economics ; St. Petersburg ; Russia
  3. Chebyshev Laboratory ; St. Petersburg State University ; St. Petersburg ; Russia
  4. I2M ; Aix-Marseille Universit茅 ; CNRS ; Marseille ; France
  
• 刊名：Geometric And Functional Analysis
• 出版年：2015
• 出版时间：April 2015
• 年：2015
• 卷：25
• 期：2
• 页码：417-452
• 全文大小：459 KB
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  3. Baranov, A., Belov, Yu., Borichev, A. (2013) Hereditary completeness for systems of exponentials and reproducing kernels. Advances in Mathematics, 235: pp. 525-554 CrossRef
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• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Analysis
  
• 出版者：Birkh盲user Basel
• ISSN：1420-8970

文摘

We solve completely the spectral synthesis problem for reproducing kernels in the de Branges spaces \({\mathcal{H}(E)}\) . Namely, we describe the de Branges spaces \({\mathcal{H}(E)}\) such that every complete and minimal system of reproducing kernels \({\{k_\lambda\}_{\lambda \in \Lambda}}\) with complete biorthogonal \({\{g_\lambda\}_{\lambda \in \Lambda}}\) admits the spectral synthesis, i.e., \({f \in \overline{\rm Span} \{(f, g_\lambda) k_\lambda : \lambda \in \Lambda\}}\) for any f in \({\mathcal{H}(E)}\) . Surprisingly, this property takes place only for two essentially different classes of de Branges spaces: spaces with finite spectral measure and spaces which are isomorphic to Fock-type spaces of entire functions. The first class goes back to de Branges himself, while special cases of de Branges spaces of the second class appeared in the literature only recently; we give a complete characterisation of this second class in terms of the spectral data for \({\mathcal{H}(E)}\) .