3.6. Redundancy in frames: What does it buy?

As illustrated by the different tables of frame bounds, frames (wavelets or windowed Fourier functions) can be very redundant (as measured by, e.g., if the frame is close to tight, and if all the frame vectors are normalized). In some applications (e.g., as in the work of the Marseille groups see the papers of Grossmann, Kronland-Martinet, Torresani) this redundancy is sought, because representations close to the continuous transform are wanted. It was noticed very early on by J. Morlet (private communication, 1986) that this redundancy also leads to robustness, in the sense that he could afford to store the wavelet coefficients ? f, ? _m,n ? with low precision (only a couple of bits), and still reconstruct f with comparatively much higher precision. Intuitively, one can understand this phenomenon as follows. Let ( ? _j) _{j ? J} be a frame (not necessarily of wavelets or windowed Fourier functions). If this frame is an orthonormal basis, then

is a unitary map, and the image of under F is all of ? ²( J). If the frame is redundant, i.e., if the ? _j are not independent, then the elements of are sequences with some correlations built into them, and is a subspace of ? ²( J), smaller than ? ²( J) itself. The more redundant the frame is, the "smaller" Ran ( F) will be.