A sharp necessary condition for rectifiable curves in metric spaces

A Necessary Condition for Weyl-Heisenberg Frames

Unions of Rectifiable Curves and the Dimension of Banach Spaces

To any metric space it is possible to associate the cardinal invariant corresponding to the least number of rectifiable curves in the space whose union is not meagre. It is shown that this invariant can vary with the metric space considered, even when restricted to the class of convex subspaces of separable Banach spaces. As a corollary it is obtained that it is consistent with set theory that ...

Some remarks about curves in metric spaces

and when this happens we say that d(x, y) is an ultrametric. One can check that an ultrametric space is totally disconnected, which is to say that it does not contain a connected subset with more than two elements. Let us say that a subset E of a metric space (M, d(x, y)) is chain connected if for every pair of points u, v ∈ E and every ǫ > 0 there is a finite chain w1, . . . , wl of points in ...

Ahlfors-regular Curves in Metric Spaces

We discuss 1-Ahlfors-regular connected sets in a general metric space and prove that such sets are ‘flat’ on most scales and in most locations. Our result is quantitative, and when combined with work of I. Hahlomaa, gives a characterization of 1-Ahlfors regular subsets of 1Ahlfors-regular curves in metric spaces. Our result is a generalization to the metric space setting of the Analyst’s (Geome...

A Sharp Sufficient Condition for Sparsity Pattern Recovery

Sufficient number of linear and noisy measurements for exact and approximate sparsity pattern/support set recovery in the high dimensional setting is derived. Although this problem as been addressed in the recent literature, there is still considerable gaps between those results and the exact limits of the perfect support set recovery. To reduce this gap, in this paper, the sufficient con...