Abstract
This chapter is primarily devoted to linear programming methods for determining bounds and exact solutions for Riesz minimal energy, best-packing, and kissing number problems on the sphere \(S^d\subset \mathbb R^{d+1}\).
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Notes
- 1.
One way to see this is to observe that each monomial \({ x}^\alpha \) in \(d+1\) variables of total degree n corresponds to a choice of d integers \(1\le b_1<\cdots <b_d\le n+d\) where \(\alpha _i=b_{i}-b_{i-1}-1\) with \(b_0=0\) and \(b_{d+1}=n+d+1\).
- 2.
Note that the dimension of a linear space of homogeneous polynomials in \(\mathbb R^{d+1}\) of the same degree has the same dimension as its restriction to \(S^d\).
- 3.
The polynomials \(Y_{nk}\) can also be defined as normalized eigenfunctions of the negative Laplace-Beltrami operator \(-\varDelta _d^*\) for \(S^d\) with eigenvalue \(\lambda _n=n(n+d-1);\) see [200].
- 4.
\(Q_n\) is also the reproducing kernel for the space \(\mathbb H_n^{d+1}\).
- 5.
We note that in the classic book of Szegő [267], \(P_n^{(\lambda )}(t)\) is used to denote \(C_n^{(\lambda )}(t)\).
- 6.
The ordering of the points in \(\omega _N\) is irrelevant in the definitions of semi-positive definite or positive definite kernel on \(S^d\).
- 7.
If x is an endpoint of I, then \(f^{(n)}(x)\) means the appropriate one-sided derivative.
- 8.
The definition of the Leech lattice is given in Section 5.8.4.
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Borodachov, S.V., Hardin, D.P., Saff, E.B. (2019). Linear Programming Bounds and Universal Optimality on the Sphere. In: Discrete Energy on Rectifiable Sets. Springer Monographs in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-0-387-84808-2_5
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DOI: https://doi.org/10.1007/978-0-387-84808-2_5
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