Hereditary completeness for systems of exponentials and reproducing kernels

2011/12/23 by Anton Baranov, Yurii Belov, Baranov, Anton +3 · 4 citations
Mathematics · #30D50 #30D55 #30H05 #46E22 #47A15 #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA) #math.CV #math.FA #msc:30D50 #msc:30D55 #msc:30H05 #msc:46E22 #msc:47A15

paper · pdf · doi:10.48550/arxiv.1112.5551

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

arxiv created 2012/03/26 · arxiv updated 2012/03/28

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^{iλ_n t}\ in L²(-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 ∑_n (f, e_n) e^{iλ_n t}, where \ e_n\ is the system biorthogonal to \e^{iλ_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.

Cited by

• On Markushevich bases \x^λ_n\_n=1^∞ for their closed span in weighted L² (A) spaces over sets A⊂ [0,∞) of positive Lebesgue measure, hereditary completeness, and moment problems
• On Strong Markushevich bases \t^λ_n\_n=1^∞ in their closed span in L² (0, 1) and characterizing a subspace of H² (\mathbbD)
• Hereditarily and nonhereditarily complete systems of vectors in a Hilbert space
• Recent developments in spectral synthesis for exponential systems and for non-self-adjoint operators

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• Hereditary completeness for systems of exponentials in weighted L²-spaces
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