Necessary and sufficient conditions to perform Spectral Tetris

Abstract

Spectral Tetris has proved to be a powerful tool for constructing sparse equal norm Hilbert space frames. We introduce a new form of Spectral Tetris which works for non-equal norm frames. It is known that this method cannot construct all frames -even in the new case introduced here. Until now, it has been a mystery as to why Spectral Tetris sometimes works and sometimes fails. We will give a complete answer to this mystery by giving necessary and sufficient conditions for Spectral Tetris to construct frames in all cases including equal norm frames, prescribed norm frames, frames with constant spectrum of the frame operator, and frames with prescribed spectrum for the frame operator. We present a variety of examples as well as special cases where Spectral Tetris always works.2000 Mathematics Subject Classification. 15B99, 42C15.

The Fundamentals of Spectral Tetris Frame Constructions

“…Next, since row one needs to square sum to 11 4 , by (3), and we only have a total weight of 1 in row one, then we need to add 11 4 − 1 = 7 4 = 1 + 3 4 ≥ 1 more weight to row one. So we will again be greedy and add another 1 in t 1,2 .…”

“…The diagonal entries of T * are now { 11 4 , 11 4 , 6 4 +?, ·}, where the third diagonal entry, and equivalently the third row, are falling short by 11 4 − 6 4 = 5 4 = 1 + 1 4 . Since 1 + 1 4 ≥ 1, then we take the eighth column of T * to be e 3 .…”

“…Theorem 6.1. [11] For M < N < 2M and λ = N M the following are equivalent: (1) The Spectral Tetris construction will successfully produce a unit norm tight…”

“…Then in Section (10), we see how the original Spectral Tetris construction method is generalized to construct 2-sparse, equidimensional, unit-weighted fusion frames. This method is further generalized in Section (11) to construct unit weighted Spectral Tetris fusion frames. Our final construction method for fusion frames occurs in Section (12) where we provide a generalized Spectral Tetris fusion frame construction algorithm as well as necessary and sufficient conditions for when this method is applicable.…”

Frame scalings: A condition number approach

“…We refer to [15] for an introduction to frame theory and to [8] for an overview of the current research in the field. Frames have traditionally played a significant role in the theory of signal processing, but today they have found application to packet based network communication [7,18], wireless sensor networks [9,10,11,12], distributed processing [7], quantum information theory, bio-medical engineering [2,25], compressed sensing [3,14], fingerprinting [26], spectral theory [6,19,20], and much more.…”