A short proof for the characterisation of tight frames

TL;DR

This paper provides a simplified proof of the fundamental theorem on the existence and characterization of finite, normalized tight frames using tensor products, and introduces tensor resolutions for symmetric positive semi-definite matrices.

Contribution

It offers a new, streamlined proof of a key theorem in frame theory and extends tensor resolution techniques to symmetric positive semi-definite matrices.

Findings

Simplified proof of Benedetto and Fickus's theorem

Tensor resolutions for symmetric positive semi-definite matrices

Enhanced understanding of tight frame characterization

Abstract

With the aid of utilising tensor products, we give a simplified proof to the fundamental theorem of Benedetto and Fickus about the existence and characterisation of finite, normalised tight frames. We also establish unit-norm tensor resolutions for symmetric, positive semi-definite matrices.