Finite tight frames and some applications

TL;DR

This paper explores finite tight frames in finite-dimensional Hilbert spaces, highlighting their advantages over orthonormal bases in symmetry, invariants, and quantization, with examples and potential applications.

Contribution

It introduces new results on integer coefficient frames and frame quantization, proposing novel applications in mathematical physics.

Findings

Finite tight frames simplify symmetry descriptions.

Frames enable new invariant forms and quantization methods.

Examples illustrate practical applications of frame theory.

Abstract

A finite-dimensional Hilbert space is usually described in terms of an orthonormal basis, but in certain approaches or applications a description in terms of a finite overcomplete system of vectors, called a finite tight frame, may offer some advantages. The use of a finite tight frame may lead to a simpler description of the symmetry transformations, to a simpler and more symmetric form of invariants or to the possibility to define new mathematical objects with physical meaning, particularly in regard with the notion of a quantization of a finite set. We present some results concerning the use of integer coefficients and frame quantization, several examples and suggest some possible applications.