Inducing Riesz and orthonormal bases in L^2 via composition operators
Yahya Saleh, Armin Iske
让 C_h 成为将 L^2(Ω_1) 映射到 L^2(Ω_2) 对于一些打开的集合 Ω_1, Ω_2 ⊆R^n 的构图运算符。 我们表征了将L^2(Ω_1)的Riesz碱基转换为L^2(Ω_2)的Riesz碱基的映射h。 将我们的分析限制在可微分映射中,我们证明保存Riesz碱基的映射h具有从零和无穷大边界的Jacobian决定因素。 我们讨论了这些结果对近似理论的影响,强调了使用双子神经网络构建具有有利近似特性的 Riesz 基的潜力。
Let C_h be a composition operator mapping L^2(Ω_1) into L^2(Ω_2) for some open sets Ω_1, Ω_2 ⊆ℝ^n. We characterize the mappings h that transform Riesz bases of L^2(Ω_1) into Riesz bases of L^2(Ω_2). Restricting our analysis to differentiable mappings, we demonstrate that mappings h that preserve Riesz bases have Jacobian determinants that are bounded away from zero and infinity. We discuss implications of these results for approximation theory, highlighting the potential of using bijective neura...