The Department of Mathematical Sciences

Spherical t-designs on the unit sphere S ⊂ R, introduced by Delsarte, Goethals, and Seidel (1977), are equal weight numerical integration rules that are exact for all polynomials of degree at most t on S. This talk considers the calculation and properties of of spherical t-designs, in particular for S where most applications reside. Bondarenko, Radchenko, and Viazovska (2013) proved that there exists a cd such that spherical t-designs withN points exist for allN ≥ cdt, which is the optimal order. Moreover they showed that there exist such spherical designs that are well-separated (2014). The interest here is in efficient spherical designs with N < t. The geometric properties of point sets on S can be characterised by their separation (twice the packing radius), their mesh norm (covering radius), and mesh ratio (covering radius / packing radius), amongst many other criteria. A common assumption arising in applications is that the the sequence of point sets is quasi-uniform, that is, their mesh ratios are uniformly bounded. The interest here is in sets of efficient spherical t-designs with small mesh ratios. Examples of spherical t-designs on S with N = t/2+O(t) points and mesh ratio < 1.8 for t = 1, . . . , 311 are available from: http://www.maths.unsw.edu.au/~rsw/Sphere/EffSphDes/ These provide excellent sets of points for both numerical integration and approximation, for example by needlets. Noon – 1:00, Wednesday, December 2, 2015. Location: KT 216 http://ipfw.edu/departments/coas/depts/math/news/seminars.html