Bounds on Discrete Potentials of Spherical (k,k)-Designs

TL;DR

This paper establishes universal bounds for potential optimization problems on spherical (k,k)-designs, demonstrating their optimality for specific potential functions and providing examples that attain these bounds.

Contribution

It derives universal bounds for polarization problems on spherical (k,k)-designs and proves their optimality for certain potential functions, expanding understanding of design optimality.

Findings

Universal bounds hold for all spherical (k,k)-designs.

Spherical (k,k)-designs are optimal for potential problems with h(t)=t^{2k}.

Examples including tight frames attain the bounds.

Abstract

We derive universal lower and upper bounds for max-min and min-max problems (also known as polarization) for the potential of spherical $(k,k)$-designs and provide certain examples, including unit-norm tight frames, that attain these bounds. The universality is understood in the sense that the bounds hold for all spherical $(k,k)$-designs and for a large class of potential functions, and the bounds involve certain nodes and weights that are independent of the potential. When the potential function is $h(t)=t^{2k}$, we prove an optimality property of the spherical $(k,k)$-designs in the class of all spherical codes of the same cardinality both for max-min and min-max potential problems.