Prime tight frames

TL;DR

This paper introduces prime tight frames, explores their properties, and shows how they can be decomposed and constructed, providing new insights and methods for analysis and synthesis in frame theory.

Contribution

It defines prime tight frames, characterizes prime harmonic tight frames, and analyzes spectral tetris constructions, advancing understanding of frame decompositions and constructions.

Findings

Any finite tight frame can be decomposed into prime tight frames.

Characterization of all prime harmonic tight frames.

Spectral tetris construction works for redundancies below two under certain conditions.

Abstract

We introduce a class of finite tight frames called prime tight frames and prove some of their elementary properties. In particular, we show that any finite tight frame can be written as a union of prime tight frames. We then characterize all prime harmonic tight frames and use this characterization to suggest effective analysis and synthesis computation strategies for such frames. Finally, we describe all prime frames constructed from the spectral tetris method, and, as a byproduct, we obtain a characterization of when the spectral tetris construction works for redundancies below two.