TY - JOUR
T1 - On spherical designs of some harmonic indices
AU - Zhu, Yan
AU - Bannai, Eiichi
AU - Bannai, Etsuko
AU - Kim, Kyoung Tark
AU - Yu, Wei Hsuan
N1 - Publisher Copyright:
© 2017, Australian National University. All rights reserved.
PY - 2017/4/13
Y1 - 2017/4/13
N2 - A finite subset Y on the unit sphere Sn−1⊆ ℝ n is called a spherical design of harmonic index t, if the following condition is satisfied: Σx∈Yf(x) = 0 for all real homogeneous harmonic polynomials f(x1, . . . , xn) of degree t. Also, for a subset T of ℕ = {1, 2, · · · }, a finite subset Y ⊆ Sn−1is called a spherical design of harmonic index T, if Σx∈Yf(x) = 0 is satisfied for all real homogeneous harmonic polynomials f(x1, . . . , xn) of degree k with k ∈ T. In the present paper we first study Fisher type lower bounds for the sizes of spherical designs of harmonic index t (or for harmonic index T). We also study ‘tight’ spherical designs of harmonic index t or index T. Here ‘tight’ means that the size of Y attains the lower bound for this Fisher type inequality. The classification problem of tight spherical designs of harmonic index t was started by Bannai-OkudaTagami (2015), and the case t = 4 was completed by Okuda-Yu (2016). In this paper we show the classification (non-existence) of tight spherical designs of harmonic index 6 and 8, as well as the asymptotic non-existence of tight spherical designs of harmonic index 2e for general e ≥ 3. We also study the existence problem for tight spherical designs of harmonic index T for some T, in particular, including index T = {8, 4}.
AB - A finite subset Y on the unit sphere Sn−1⊆ ℝ n is called a spherical design of harmonic index t, if the following condition is satisfied: Σx∈Yf(x) = 0 for all real homogeneous harmonic polynomials f(x1, . . . , xn) of degree t. Also, for a subset T of ℕ = {1, 2, · · · }, a finite subset Y ⊆ Sn−1is called a spherical design of harmonic index T, if Σx∈Yf(x) = 0 is satisfied for all real homogeneous harmonic polynomials f(x1, . . . , xn) of degree k with k ∈ T. In the present paper we first study Fisher type lower bounds for the sizes of spherical designs of harmonic index t (or for harmonic index T). We also study ‘tight’ spherical designs of harmonic index t or index T. Here ‘tight’ means that the size of Y attains the lower bound for this Fisher type inequality. The classification problem of tight spherical designs of harmonic index t was started by Bannai-OkudaTagami (2015), and the case t = 4 was completed by Okuda-Yu (2016). In this paper we show the classification (non-existence) of tight spherical designs of harmonic index 6 and 8, as well as the asymptotic non-existence of tight spherical designs of harmonic index 2e for general e ≥ 3. We also study the existence problem for tight spherical designs of harmonic index T for some T, in particular, including index T = {8, 4}.
KW - Delsarte’s method
KW - Elliptic diophantine equation
KW - Fisher type lower bound
KW - Gegenbauer polynomial
KW - Larman-Rogers-Seidel’s theorem
KW - Semidefinite programming
KW - Spherical designs of harmonic index
KW - Tight design
UR - http://www.scopus.com/inward/record.url?scp=85018508280&partnerID=8YFLogxK
U2 - 10.37236/6437
DO - 10.37236/6437
M3 - 期刊論文
AN - SCOPUS:85018508280
SN - 1077-8926
VL - 24
JO - Electronic Journal of Combinatorics
JF - Electronic Journal of Combinatorics
IS - 2
ER -