Null spherical t-designs: Some considerations - Campena, Francis Joseph H. (Animo Repository, 2012)
This paper is based on the lectures of Professor Eiichi Bannai during the Spring of 2012 at Shanghai Jiao Tong University, Shanghai China. In the said lectures, the idea of spherical t-designs were presented and Prof. Eiichi introduced the concept of null spherical t-designs as the analogue of the combatorial null t-designs. It is defined as a finite set X of points on the unit sphere in n dimension together with a weight function w: X ↦ R - {10} such that the following condition is satisfied Σ w (x) f (x) = 0 x∈X for any polynomial of degree at most t.Similar to the problems in spherical t-designs, explicit construction of such designs are difficult aside from a few known that exists. This paper provides some constructions of null spherical t-designs based on the existence of spherical t-designs and cubature formulas on the sphere. We also provide some basic properties of null spherical t-designs similar to the properties of spherical t-designs.