From Sidelnikov-Welch bounds to projection constants

Abstract

In the paper, we prove a recursive version of the weighted Sidelnikov-Welch inequality for real and complex unit vectors. Unlike the classical form, which gives a direct lower bound for a fixed even power sum, our inequality relates two consecutive even power sums. Iteration yields the usual weighted Sidelnikov-Welch bound. We apply this estimate to maximal relative projection constants. If ${\mathbb{K}}^m$ admits a maximal equiangular tight frame with $M_{\mathbb{K}}$ vectors, then for every integer $k\ge 1$, $${\lambda }_{\mathbb{K}}(k{M}_{\mathbb{K}}-m,k{M}_{\mathbb{K}})={\lambda }_{\mathbb{K}}(m)-\frac{2m}{k{M}_{\mathbb{K}}}+1.$$ Moreover, the maximal value is realized by an equiangular tight frame when $k=1$ and by a biangular tight frame when $k\ge 2$.