Marjorie K. Drake | Searcharxiv

arXiv2024

Finiteness Principles for Smooth Convex Functions

Let $E \subset \mathbb{R}^n$ be a compact set, and $f:E \to \mathbb{R}$. How can we tell if there exists a convex extension $F \in C^{1,1}(\mathbb{R}^n)$ of $f$, i.e. satisfying $F|_E = f|_E$? Assuming such an extension exists, how small can one take the Lipschitz constant $\text{Lip}(\nabla F): = \sup_{x,y \in \mathbb{R}^n, x \neq y} \frac{|\nabla F(x) - \nabla F(y)|}{|x-y|}$? We provide an answer to these questions for the class of strongly convex functions by proving that there exist constants $k^\# \in \mathbb{N}$ and $C>0$ depending only on the dimension $n$, such that if for every subset $S \subset E$, $\#S \leq k^\#$, there exists an $η$-strongly convex function $F^S \in C^{1,1}(\mathbb{R}^n)$ satisfying $F^S|_S=f|_S$ and $\text{Lip}(\nabla F^S) \leq M$, then there exists an ${\fracη{C}}$-strongly convex function $F \in C^{1,1}_c(\mathbb{R}^n)$ satisfying $F|_E = f|_E$, and $\text{Lip}(\nabla F) \leq C M^2/η$. Further, we prove a Finiteness Principle for the space of convex functions in $C^{1,1}(\mathbb{R})$ and that the sharp finiteness constant for this space is $k^\#=5$.

math.CA↗

arXiv2022

Approximate Extension in Sobolev Space

Let $L^{m,p}(\mathbb{R}^n)$ be the homogeneous Sobolev space for $p \in (n,\infty)$, $μ$ be a Borel regular measure on $\mathbb{R}^n$, and $L^{m,p}(\mathbb{R}^n) + L^p(dμ)$ be the space of Borel measurable functions with finite seminorm $\|f\|_{L^{m,p}(\mathbb{R}^n) + L^p(dμ)} := \text{inf}_{f_1 +f_2 = f} \{ \|f_1\|_{L^{m,p}(\mathbb{R}^n)}^p + \int_{\mathbb{R}^n} |f_2|^p dμ\}^{1/p}$. We construct a linear operator $T:L^{m,p}(\mathbb{R}^n) + L^p(dμ) \to L^{m,p}(\mathbb{R}^n)$, that nearly optimally decomposes every function in the sum space: $\|Tf\|_{L^{m,p}(\mathbb{R}^n)}^p + \int_{\mathbb{R}^n} |Tf-f|^p dμ\leq C \|f\|_{L^{m,p}(\mathbb{R}^n) + L^p(dμ)}^p$ with $C$ dependent on $m$, $n$, and $p$ only. For $E \subset \mathbb{R}^n$, let $L^{m,p}(E)$ denote the space of all restrictions to $E$ of functions $F \in L^{m,p}(\mathbb{R}^n)$, equipped with the standard trace seminorm. For $p \in (n, \infty)$, we construct a linear extension operator $T:L^{m,p}(E) \to L^{m,p}(\mathbb{R}^n)$ satisfying $Tf|_E = f|_E$ and $\|Tf\|_{L^{m,p}(\mathbb{R}^n)} \leq C \|f\|_{L^{m,p}(E)}$, where $C$ depends only on $n$, $m$, and $p$. We show these operators can be expressed through a collection of linear functionals whose supports have bounded overlap.

math.FA↗