Necessary and sufficient conditions to perform Spectral Tetris

TL;DR

This paper provides a comprehensive set of necessary and sufficient conditions for when Spectral Tetris can successfully construct various types of frames, including equal norm, prescribed norm, and specific spectrum frames.

Contribution

It introduces a new form of Spectral Tetris for non-equal norm frames and fully explains the conditions under which it succeeds or fails.

Findings

Complete characterization of Spectral Tetris success conditions

Extension of Spectral Tetris to non-equal norm frames

Identification of cases where Spectral Tetris always works

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.