This talk presents a new constructive method for the efficient subsampling of finite frames in C^m. Based on a suitable random subsampling strategy, it is possible to extract from any given frame with bounds 0<A≤B<∞ (and condition B/A) a similarly conditioned reweighted subframe consisting of merely O(m\log m) elements. Further, utilizing a deterministic subsampling method based on principles developed in by Batson, Spielman, and Srivastava, we are able to reduce the number of elements to O(m) (with a constant close to one). By controlling the weights via a preconditioning step, we can, in addition, preserve the lower frame bound in the unweighted case. This allows to derive new quasi-optimal unweighted (left) Marcinkiewicz-Zygmund inequalities for L²(D, μ) with constructible node sets of size O(m) for m-dimensional subspaces of bounded functions. Those can be applied e.g. for (plain) least-squares sampling reconstruction of functions, where we obtain new quasi-optimal results avoiding the Kadison-Singer theorem.

This is joint work with Martin Schäfer and Tino Ullrich.