核心概念
When two Lip(γ) functions are close on a subset B, they remain close in a weaker Lip(η) sense throughout the larger set Σ, provided η < γ.
要約
The key insights and highlights of the content are:
The article investigates the consequences when two Lip(γ) functions, in the sense of Stein, are close throughout a subset B of their domain Σ.
The main result is the Lipschitz Sandwich Theorem 3.1, which states that if ψ and ϕ are Lip(γ) functions on Σ with bounded Lip(γ) norms, and ψ and ϕ coincide on a closed subset B that is a δ0-cover of Σ, then the Lip(η) norm of ψ - ϕ is bounded above by ε throughout Σ, provided η < γ.
The theorem requires that the closeness of ψ and ϕ on B is measured in a pointwise sense, rather than requiring the stronger condition that the Lip(γ) norm of ψ - ϕ on B is bounded.
The restriction that η < γ is shown to be sharp, as the result is false when η = γ.
The Lipschitz Sandwich Theorem has applications in cost-effective approximation of Lip(γ) functions, as shown in Section 4.
The article also establishes related results, such as the Single-Point Lipschitz Sandwich Theorem 3.7 and the Pointwise Lipschitz Sandwich Theorem 3.9, which consider the case when the subset B is a single point.
The proofs of the main results rely on careful estimates involving the Lip(γ) structure of the functions, as detailed in Sections 5-10.