Analysis reveals composition operators transform Riesz bases in L2 spaces, suggesting neural networks enhance approximation properties.
Let Cₕ C h be a composition operator mapping L²(Ω ₁) L 2 ( Ω 1 ) into L²(Ω ₂) L 2 ( Ω 2 ) for some open sets Ω ₁, Ω ₂ ⊆ Rⁿ Ω 1 , Ω 2 ⊆ R n . We characterize the mappings h that transform Riesz bases of L²(Ω ₁) L 2 ( Ω 1 ) into Riesz bases of L²(Ω ₂) 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 neural networks to construct Riesz bases with favorable approximation properties.
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Saleh et al. (2025) studied this question.
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