Construction of spherical cubature formulas using lattices

Cubature formulas, geometrical designs, reproducing kernels, and Markov operators

Cubature formulas and geometrical designs are described in terms of reproducing kernels for Hilbert spaces of functions on the one hand, and Markov operators associated to orthogonal group representations on the other hand. In this way, several known results for spheres in Euclidean spaces, involving cubature formulas for polynomial functions and spherical designs, are shown to generalize to large classes of finite measure spaces $(Ω,σ)$ and appropriate spaces of functions inside $L^2(Ω,σ)$. The last section points out how spherical designs are related to a class of reflection groups which are…

Integral Representation and Computation a Multiple Sum in the Theory of Cubature Formulas

In this article we furnish a new simple proof of a hard identity from the theory of cubature formulas via the method of coefficients.

Lower bounds for projective designs, cubature formulas and related isometric embeddings

Yudin's lower bound for the spherical designs is generalized to the cubature formulas on the projective spaces over a field K, where K can be R, C, or H (the field of quaternions), and thus to isometric embeddings of l_2 into l_p with p an even integer. For large p and in some other situations this is essentially better than those known before.

Cubature formulas and Sobolev inequalities

We study a problem in the theory of cubature formulas on the sphere: given $θ\in (0, 1)$, determine the infimum of $\|ν\|_θ= \sum_{i = 1}^n ν_i^θ$ over cubature formulas $ν$ of strength $t$, where $ν_i$ are the weights of the formula $ν$. This problem, which generalizes the classical problem of bounding the minimal cardinality of a cubature formula -- the case $θ= 0$ -- was introduced in recent work of Hang and Wang (arXiv:2010.10654), who showed the problem to be related to optimal constants in Sobolev inequalities. Using the elementary theory of reproducing kernel Hilbert spaces on $S^{n - 1…

基于Larman-Rogers-Seidel定理的最小立方公式存在性问题的新方法

A new approach for the existence problem of minimal cubature formulas based on the Larman-Rogers-Seidel theorem

针对球对称积分，研究在Möller下界取等号的4k+1次立方公式的存在性，证明高维最小公式中同心球面上任意两点的内积均为有理数，并由此证明对某些特殊积分，d>2时不存在13次和21次的最小公式。

关于从群轨道得到的求积公式和设计的注记

Note on cubature formulae and designs obtained from group orbits

本文基于Sobolev定理，给出Xu定理的简化证明，并将Neumaier-Seidel定理推广到不变欧几里得设计，从而分类由角向量轨道并集得到的紧欧几里得设计。