Pete Casazza, Piecewise Scalable Frames, 2022.09.27

Speaker: Pete Casazza (University of Missouri)
Title: Piecewise Scalable Frames
Date: 09.27.2022

Abstract: A Hilbert space frame {x_i}_{i=1}^{m} for R^n is scalable if there exist constants {a_i}_{i=1}^{m} so that {a_i x_i}_{i=1}^{m} is a Parseval frame. I.e., for every x in R^n,
x=\sum_{i=1}^{m}⟨x,a_i x_i⟩a_i x_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 R^2 and 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_i Px_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 R^2 and 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.

Видео Pete Casazza, Piecewise Scalable Frames, 2022.09.27 канала CodEx Seminar