Spherical designs of harmonic index t

TL;DR

This paper introduces spherical designs of harmonic index t, providing their construction, a lower bound on their size, and exploring the existence of designs that attain this bound.

Contribution

It defines spherical designs of harmonic index t, establishes a Fisher type lower bound on their size, and investigates the existence of optimal designs.

Findings

Constructed spherical designs of harmonic index t.

Derived a Fisher type lower bound on the size.

Explored the existence of designs attaining the bound.

Abstract

Spherical $t$-design is a finite subset on sphere such that, for any polynomial of degree at most $t$, the average value of the integral on sphere can be replaced by the average value at the finite subset. It is well-known that an equivalent condition of spherical design is given in terms of harmonic polynomials. In this paper, we define a spherical design of harmonic index $t$ from the viewpoint of this equivalent condition, and we give its construction and a Fisher type lower bound on the cardinality. Also we investigate whether there is a spherical design of harmonic index attaining the bound.