The Advent of

edundancy is a com-mon tool in our daily lives. Before we leave the house, we double- and triple-check that we turned off gas and lights, took our keys, and have money (at least those worrywarts among us do). When an important date is coming up, we drive our loved ones crazy by confirming “just once more ” they are on top of it. Of course, the reason we are doing that is to avoid a disaster by missing or forgetting something, not to drive our loved ones crazy. The same idea of removing doubt is present in signal representations. Given a signal, we represent it in anoth-er system, typically a basis, where its characteristics are more readily apparent in the transform coefficients. However, these representations are typically nonredundant, and thus corruption or loss of transform coefficients can be serious. In comes redundancy; we build a safety net into our representation so that we can avoid those dis-asters. The redundant counterpart of a basis is called a frame [no one seems to know why they are called frames, perhaps because of the bounds in (25)?]. It is generally acknowledged (at least in the signal processing and harmonic analysis communities) that frames were born in 1952 in the paper by Duffin and Schaeffer [32]. Despite being over half a century old, frames gained popularity only in the last decade, due mostly to the work of the three wavelet pioneers—Daubechies, Grossman, and Meyer [29]. Frame-like ideas, that is, building redundancy into a signal expansion, can be found in pyrami