Null spherical t-designs

• Main Author: Campena, Francis Joseph H.
• Format: text
• Language: English
• Published: Animo Repository 2016
• Subjects: Combinatorial designs and configurations; Discrete Mathematics and Combinatorics; Mathematics
• Online Access: https://animorepository.dlsu.edu.ph/etd_doctoral/475
• Institution: De La Salle University

The survey paper [2] of Eiichi Bannai and Etsuko Bannai provided an overview of the study of spherical designs and algebraic combinarotics. In the survey paper the authors focused on the study of "good" finite subsets of the unit sphere in n-dimension, n{u100000}1 and that part of their problem is to define what "good finite subsets" should mean. However, up to today, no definite answer is known and it is unrealistic to expect a single good answer. A possible point of view that one could take is to de ne a good subset of the unit sphere to be the one that globally approximates the whole sphere using only a finite number of point. A reasonable definition to what it means for a finite subset to approximate the sphere was given by Delsarte-Goethals-Seidel in 1966 as follows: a finite subset X on n{u100000}1 is called a spherical t-design on n{u100000}1, if for any polynomial f(x) = f(x1 x2 : : : xn) of degree at most t, the value of the integral of f(x) on n{u100000}1 (divided by the volume of n{u100000}1) is just the average value of f(x) on the finite set X that is, 1 j n{u100000}1j Z x2 n{u100000}1 f(x)d (x) = 1 jX j X x2X f(x) where is a Lesbegue measure on n{u100000}1: In In one of the talks on Algebraic Combinatorics at Shanghai Jiao Tong University on May 2012, Eiichi Bannai defined the notion of a null spherical t-design on the unit sphere in n-dimension. For any non-negative integers n t such that n > 1 and t 0 a pair (X !) is a null spherical t-design on n{u100000}1 if X is a finite subset of n{u100000}1 and ! is a non-zero weight function on X that satisfies X x2X !(x)f(x) = 0 for any homogeneous harmonic polynomial f(x) in n variables of degree at most t: This definition generalizes the notion of the usual spherical t-designs on the unit sphere by allowing non-zero weights. In this study, properties of null spherical t-designs similar to properties of spherical t-designs are presented. Construction of null spherical designs is also provided using known spherical designs. Null spherical designs are also described using the Gegenbauer polynomials and characteristic matrices. Bounds on the number of points in a null spherical design are determined. In particular, we conjecture that the minimum number of points in a null spherical t-design on n{u100000}1 is 2(t + 1):