| Abstrakt: |
A de Branges space $ \mathcal {B} $ B is regular if the constants belong to its space of associated functions and is symmetric if it is isometrically invariant under the map $ F(z) \mapsto F(-z) $ F (z) ↦ F (− z). Let $ K_\mathcal {B}(z,w) $ K B (z , w) be the reproducing kernel in $ \mathcal {B} $ B and $ S_\mathcal {B} $ S B be the operator of multiplication by the independent variable with maximal domain in $ \mathcal {B} $ B . Loosely speaking, we say that $ \mathcal {B} $ B has the $ \ell _p $ ℓ p -oversampling property relative to a proper subspace $ \mathcal {A} $ A of it, with $ p\in (2,\infty ] $ p ∈ (2 , ∞ ] , if there exists $ J_{\mathcal {A}\mathcal {B}}:{\mathbb C}\times {\mathbb C}\to {\mathbb C} $ J A B : C × C → C such that $ J(\cdot,w)\in \mathcal {B} $ J (⋅ , w) ∈ B for all $ w\in {\mathbb C} $ w ∈ C , $$\begin{align*} &\sum_{\lambda\in\sigma(S_{\mathcal{B}}^{\gamma})} \left(\frac{\left\lvert J_{\mathcal{A}\mathcal{B}}(z,\lambda) \right\rvert}{K_\mathcal{B}(\lambda,\lambda)^{1/2}}\right)^{p/(p-1)} ∑ λ ∈ σ (S B γ) (| J A B (z , λ) | K B (λ , λ) 1 / 2 ) p / (p − 1) < ∞ and F (z) = ∑ λ ∈ σ (S B γ) J A B (z , λ) K B (λ , λ) F (λ) , for all $ F\in \mathcal {A} $ F ∈ A and almost every self-adjoint extension $ S_{\mathcal {B}}^{\gamma } $ S B γ of $ S_\mathcal {B} $ S B . This definition is motivated by the well-known oversampling property of Paley-Wiener spaces. In this paper, we provide sufficient conditions for a symmetric, regular de Branges space to have the $ \ell _p $ ℓ p -oversampling property relative to a chain of de Branges subspaces of it. [ABSTRACT FROM AUTHOR] |
