TY - JOUR
T1 - Discrepancy, separation and Riesz energy of finite point sets on the unit sphere
AU - Leopardi, Paul
PY - 2013/7
Y1 - 2013/7
N2 - For d ≥ 2 we consider asymptotically equidistributed sequences of Sd codes, with an upper bound δ on spherical cap discrepancy, and a lower bound Δ on separation. For such sequences, if 0 < s < d, then the difference between the normalized Riesz s energy of each code, and the normalized s-energy double integral on the sphere is bounded above by O(δ1-s/dΔ-sN-s/d where N is the number of code points. For well separated sequences of spherical codes, this bound becomes δ1-s/d We apply these bounds to minimum energy sequences, sequences of well separated spherical designs, sequences of extremal fundamental systems, and sequences of equal area points.
AB - For d ≥ 2 we consider asymptotically equidistributed sequences of Sd codes, with an upper bound δ on spherical cap discrepancy, and a lower bound Δ on separation. For such sequences, if 0 < s < d, then the difference between the normalized Riesz s energy of each code, and the normalized s-energy double integral on the sphere is bounded above by O(δ1-s/dΔ-sN-s/d where N is the number of code points. For well separated sequences of spherical codes, this bound becomes δ1-s/d We apply these bounds to minimum energy sequences, sequences of well separated spherical designs, sequences of extremal fundamental systems, and sequences of equal area points.
KW - Riesz energy
KW - Separation
KW - Sphere
KW - Spherical cap discrepancy
UR - http://www.scopus.com/inward/record.url?scp=84879249137&partnerID=8YFLogxK
U2 - 10.1007/s10444-011-9266-4
DO - 10.1007/s10444-011-9266-4
M3 - Article
SN - 1019-7168
VL - 39
SP - 27
EP - 43
JO - Advances in Computational Mathematics
JF - Advances in Computational Mathematics
IS - 1
ER -