Codes and Expansions (CodEx) Seminar

Kasso A. Okoudjou (Tufts University)
Minimizing the \(p\) frame potentials

Given \( d \geq 2\), \( p \in (0,\infty] \), and \( N \geq 2 \), the discrete \(p\) frame potential is defined by
\( FP_{p,d,N}(\Phi) := \sum_{k,\ell=1}^N | \langle \varphi_k, \varphi_\ell \rangle |^p \)
where \( \Phi = \{ \varphi_k \}_{k=1}^N \subset S^{d-1} \), and \( S^{d-1} \) is the unit sphere in \( \mathbb{R}^d \). This function can be viewed as the restriction to discrete probability measures, of the continuous \(p\)-frame potentials defined by
\( \iint_{S^{d-1} \times S^{d-1}} | \langle x, y \rangle |^p d\mu(x) d\mu(y) \)
where \( \mu \) runs over the set of all Borel probability measures defined on \( S^{d-1} \). In this talk, we will focus on the relation between the minimizers of these two functionals. In particular, we will report on ongoing work in dimension \( d = 2 \) where the classification of the minimizers of these functionals is still incomplete.