Codes and Expansions (CodEx) Seminar

Pete Casazza (University of Missouri)
Piecewise Scalable Frames

A Hilbert space frame \(\{x_i\}_{i=1}^m\) for \(\mathbb{R}^n\) is scalable if there exist constants \(\{a_i\}_{i=1}^m\) so that \(\{a_ix_i\}_{i=1}^m\) is a Parseval frame. I.e. For every \(x\in\mathbb{R}^n\), \(x=\sum_{i=1}^m\langle x,a_ix_i\rangle a_ix_i\). Scalable frames are very useful in applications since they reduce exponentially the number of calculations needed to carry out the application. So there is a large amount of literature on this topic. But this topic has two major drawbacks:
(1) Very few frames are actually scalable - even in \(\mathbb{R}^2\) and \(\mathbb{R}^3\).
(2) To scale a frame, most of the vectors are thrown away and the remaining ones are scaled - so they are useless in applications.
We introduce a generalization of scalable frames we call piecewise scalable frames when there exists a projection \(P\) on the space and constants \(\{a_i, b_i\}_{i=1}^m\) so that \(\{a_iP x_i + b_i(I - P)x_i\}_{i=1}^m\) is a Parseval frame. Piecewise scalable frames take care of many of the problems with scalable frames:
(1) For example, all frames in \(\mathbb{R}^2\) and \(\mathbb{R}^3\) are piecewise scalable.
(2) Often, we can piecewise scale a frame without having to set any vector equal to zero.
We will go over the basic properties of piecewise scalable frames.