Let <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$C_h$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>C</mi> <mi>h</mi> </msub> </math></EquationSource> </InlineEquation> be a composition operator mapping <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$L^2(\Omega _1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">Ω</mi> <mn>1</mn> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> into <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$L^2(\Omega _2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">Ω</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for some open sets <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\Omega _1, \Omega _2 \subseteq {\mathbb {R}}^n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Ω</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi mathvariant="normal">Ω</mi> <mn>2</mn> </msub> <mo>⊆</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>. We characterize the mappings h that transform Riesz bases of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$L^2(\Omega _1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">Ω</mi> <mn>1</mn> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> into Riesz bases of <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$L^2(\Omega _2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">Ω</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. 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.

Inducing Riesz Bases in $L^2$ via Composition Operators

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2025年12月3日

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Yahya Saleh,
Armin Iske

摘要

Let $C_h$ $C_{h}$ be a composition operator mapping $L^2(\Omega _1)$ $L^{2} (Ω_{1})$ into $L^2(\Omega _2)$ $L^{2} (Ω_{2})$ for some open sets $\Omega _1, \Omega _2 \subseteq {\mathbb {R}}^n$ $Ω_{1}, Ω_{2} \subseteq R^{n}$ . We characterize the mappings h that transform Riesz bases of $L^2(\Omega _1)$ $L^{2} (Ω_{1})$ into Riesz bases of $L^2(\Omega _2)$ $L^{2} (Ω_{2})$ . 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.

Inducing Riesz Bases in \(L^2\) via Composition Operators

摘要