The Distributions of Hilbert Space Frame Vectors and Frame Coefficients

TL;DR

This paper investigates the distribution of frame coefficients and related distances in Hilbert space, providing exact calculations for special frames and highlighting a neglected area of study in frame theory.

Contribution

It offers the first detailed analysis of coefficient distributions in Hilbert space frames, including exact results for specific frame types.

Findings

Exact distribution formulas for unit norm tight frames

Distribution analysis for equiangular frames

Identification of a previously overlooked research area

Abstract

The most fundamental notion for Hilbert space frames is the sequence of frame coefficients for a vector x in the space. Yet, we know little about the distribution of these coefficient sequences. In this paper, we make the first detailed study of the distribution of the frame coefficients for vectors in a Hilbert space, as well as the related notion of the square sums of the distances of vectors in the space from the frame vectors. We will give some surprisingly exact calculations for special frames such as unit norm tight frames and equiangular frames. This is a study which should have been done 20 years ago.