Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userLinear Lipschitz and $C^1$ extension operators through random projection
68
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We construct a regular random projection of a metric space onto a closed doubling subset and use it to linearly extend Lipschitz and $C^1$ functions. This way we prove more directly a result by Lee and Naor and we generalize the $C^1$ extension theorem by Whitney to Banach spaces.
rate research
Read More
Fix integers $m geq 2$, $n geq 1$. Let $C^{m-1,1}(mathbb{R}^n)$ be the space of $(m-1)$-times differentiable functions $F : mathbb{R}^n rightarrow mathbb{R}$ whose $(m-1)$st order partial derivatives are Lipschitz continuous, equipped with a standard seminorm. Given $E subseteq mathbb{R}^n$, let $C^{m-1,1}(E)$ be the trace space of all restrictions $F|_E$ of functions $F$ in $C^{m-1,1}(mathbb{R}^n)$, equipped with the standard quotient (trace) seminorm. We prove that there exists a bounded linear operator $T : C^{m-1,1}(E) rightarrow C^{m-1,1}(mathbb{R}^n)$ satisfying $Tf|_E = f$ for all $f in C^{m-1,1}(E)$, with operator norm at most $exp( gamma D^k)$, where $D := binom{m+n-1}{n}$ is the number of multiindices of length $n$ and order at most $m-1$, and $gamma,k > 0$ are absolute constants (independent of $m,n,E$). Our results improve on the previous construction of linear extension operators with norm at most $ exp(gamma D^k 2^D)$.
We study vertex cut and flow sparsifiers that were recently introduced by Moitra, and Leighton and Moitra. We improve and generalize their results. We give a new polynomial-time algorithm for constructing O(log k / log log k) cut and flow sparsifiers, matching the best existential upper bound on the quality of a sparsifier, and improving the previous algorithmic upper bound of O(log^2 k / log log k). We show that flow sparsifiers can be obtained from linear operators approximating minimum metric extensions. We introduce the notion of (linear) metric extension operators, prove that they exist, and give an exact polynomial-time algorithm for finding optimal operators. We then establish a direct connection between flow and cut sparsifiers and Lipschitz extendability of maps in Banach spaces, a notion studied in functional analysis since 1930s. Using this connection, we prove a lower bound of Omega(sqrt{log k/log log k}) for flow sparsifiers and a lower bound of Omega(sqrt{log k}/log log k) for cut sparsifiers. We show that if a certain open question posed by Ball in 1992 has a positive answer, then there exist tilde O(sqrt{log k}) cut sparsifiers. On the other hand, any lower bound on cut sparsifiers better than tilde Omega(sqrt{log k}) would imply a negative answer to this question.
For suitable finite-dimensional smooth manifolds M (possibly with various kinds of boundary or corners), locally convex topological vector spaces F and non-negative integers k, we construct continuous linear operators S_n from the space of F-valued k times continuously differentiable functions on M to the corresponding space of smooth functions such that S_n(f) converges to f in C^k(M,F) as n tends to infinity, uniformly for f in compact subsets of C^k(M,F). We also study the existence of continuous linear right inverses for restriction maps from C^k(M,F) to C^k(L,F) if L is a closed subset of M, endowed with a C^k-manifold structure turning the inclusion map from L to M into a C^k-map. Moreover, we construct continuous linear right inverses for restriction operators between spaces of sections in vector bundles in many situations, and smooth local right inverses for restriction operators between manifolds of mappings. We also obtain smoothing results for sections in fibre bundles.
We prove that a bi-Lipschitz image of a planar $BV$-extension domain is also a $BV$-extension domain, and that a bi-Lipschitz image of a planar $W^{1,1}$-extension domain is again a $W^{1,1}$-extension domain.
A well-known result says that the Euclidean unit ball is the unique fixed point of the polarity operator. This result implies that if, in $mathbb{R}^n$, the unit ball of some norm is equal to the unit ball of the dual norm, then the norm must be Euclidean. Motivated by these results and by relatively recent results in convex analysis and convex geometry regarding various properties of order reversing operators, we consider, in a real Hilbert space setting, a more general fixed point equation in which the polarity operator is composed with a continuous invertible linear operator. We show that if the linear operator is positive definite, then the considered equation is uniquely solvable by an ellipsoid. Otherwise, the equation can have several (possibly infinitely many) solutions or no solution at all. Our analysis yields a few by-products of possible independent interest, among them results related to coercive bilinear forms (essentially a quantitative convex analytic converse to the celebrated Lax-Milgram theorem from partial differential equations) and a characterization of real Hilbertian spaces.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
