Geometry of the Welch Bounds

TL;DR

This paper presents a geometric approach to deriving the Welch bounds, unifying various observations about tightness, frames, and t-designs, and extends the bounds to generalized and continuous frames.

Contribution

It introduces a geometric perspective that unifies the derivation of Welch bounds and connects them to symmetric k-tensors, tight frames, and t-designs, extending to generalized frames.

Findings

Derived the entire family of Welch bounds using geometric methods

Connected tightness of bounds to the existence of tight frames

Extended Welch bounds to generalized and continuous frames

Abstract

A geometric perspective involving Grammian and frame operators is used to derive the entire family of Welch bounds. This perspective unifies a number of observations that have been made regarding tightness of the bounds and their connections to symmetric k-tensors, tight frames, homogeneous polynomials, and t-designs. In particular. a connection has been drawn between sampling of homogeneous polynomials and frames of symmetric k-tensors. It is also shown that tightness of the bounds requires tight frames. The lack of tight frames in symmetric k-tensors in many cases, however, leads to consideration of sets that come as close as possible to attaining the bounds. The geometric derivation is then extended in the setting of generalized or continuous frames. The Welch bounds for finite sets and countably infinite sets become special cases of this general setting.