2025/05/30 by Elias Zikkos, Zikkos, Elias
Mathematics · Physics and Astronomy · #30B50 #30B60 #47A10 #Advanced Differential Equations and Dynamical Systems #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA) #Nonlinear Waves and Solitons #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2505.24761
openalex publication_date 2025/05/30 · openalex created_date 2025/10/06 · openalex updated_date 2026/08/02
Let Λ=\λn\n=1∞ be a strictly increasing sequence of positive real numbers such that ∑n=1∞(1)/(λn)<∞ and inf(λn+1-λn)>0. We investigate properties of the closed span of the system \tλn\n=1∞ in L2 (0,1), denoted by MΛ, and of the unique biorthogonal family \rn (t)\n=1∞ to the system \tλn\n=1∞ in MΛ. We show that the system \tλn\n=1∞ is a strong Markushevich basis in MΛ and we obtain a series representation for functions in MΛ. We also construct a general class of operators on MΛ that admit spectral synthesis. In particular, for all ρ∈ (0,1) the operator Tρ(f)=f(ρx) on MΛ admits spectral synthesis. In addition, we characterize a certain subspace of the classical Hardy space H2 (\mathbbD). Under the extra assumption that Λ⊂ℕ, let H2(\mathbbD, Λ) consist of functions f in H2(\mathbbD) so that the Fourier coefficients cn of the boundary function f(eiθ) vanish for all n∉ Λ. We prove that f∈ H2(\mathbbD, Λ) if and only if f∈MΛ and ∑n=1∞| ⟨ f, rn⟩ |2<∞, where ⟨ f, g⟩= ∫01 f(t)⋅ g(t) dt.