Higher Order Lipschitz Sandwich Theorems
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| author | Lyons, Terry McLeod, Andrew D. |
| author_facet | Lyons, Terry McLeod, Andrew D. |
| contents | We investigate the consequence of two Lip$(γ)$ functions, in the sense of Stein, being close throughout a subset of their domain. A particular consequence of our results is the following. Given $K_0 > \varepsilon > 0$ and $γ> η> 0$ there is a constant $δ= δ(γ,η,\varepsilon,K_0) > 0$ for which the following is true. Let $Σ\subset \mathbb{R}^d$ be closed and $f , h : Σ\to \mathbb{R}$ be Lip$(γ)$ functions whose Lip$(γ)$ norms are both bounded above by $K_0$. Suppose $B \subset Σ$ is closed and that $f$ and $h$ coincide throughout $B$. Then over the set of points in $Σ$ whose distance to $B$ is at most $δ$ we have that the Lip$(η)$ norm of the difference $f-h$ is bounded above by $\varepsilon$. More generally, we establish that this phenomenon remains valid in a less restrictive Banach space setting under the weaker hypothesis that the two Lip$(γ)$ functions $f$ and $h$ are only close in a pointwise sense throughout the closed subset $B$. We require only that the subset $Σ$ be closed; in particular, the case that $Σ$ is finite is covered by our results. The restriction that $η< γ$ is sharp in the sense that our result is false for $η:= γ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_06849 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Higher Order Lipschitz Sandwich Theorems Lyons, Terry McLeod, Andrew D. Classical Analysis and ODEs Numerical Analysis Differential Geometry 26B35, 26D07, 46A32, 46B28, 46M05 We investigate the consequence of two Lip$(γ)$ functions, in the sense of Stein, being close throughout a subset of their domain. A particular consequence of our results is the following. Given $K_0 > \varepsilon > 0$ and $γ> η> 0$ there is a constant $δ= δ(γ,η,\varepsilon,K_0) > 0$ for which the following is true. Let $Σ\subset \mathbb{R}^d$ be closed and $f , h : Σ\to \mathbb{R}$ be Lip$(γ)$ functions whose Lip$(γ)$ norms are both bounded above by $K_0$. Suppose $B \subset Σ$ is closed and that $f$ and $h$ coincide throughout $B$. Then over the set of points in $Σ$ whose distance to $B$ is at most $δ$ we have that the Lip$(η)$ norm of the difference $f-h$ is bounded above by $\varepsilon$. More generally, we establish that this phenomenon remains valid in a less restrictive Banach space setting under the weaker hypothesis that the two Lip$(γ)$ functions $f$ and $h$ are only close in a pointwise sense throughout the closed subset $B$. We require only that the subset $Σ$ be closed; in particular, the case that $Σ$ is finite is covered by our results. The restriction that $η< γ$ is sharp in the sense that our result is false for $η:= γ$. |
| title | Higher Order Lipschitz Sandwich Theorems |
| topic | Classical Analysis and ODEs Numerical Analysis Differential Geometry 26B35, 26D07, 46A32, 46B28, 46M05 |
| url | https://arxiv.org/abs/2404.06849 |