Construction of spherical \(t\)-designs
Let \(d\) and \(t\) be positive integers and let \(S^{d-1}\) be the unit sphere in \(\mathbb{R}^ d\). \textit{P. Delsarte}, \textit{J. M. Goethals} and \textit{J. J. Seidel} [Geom. Dedicata, 6, 363-388 (1977; Zbl 0376.05015)] introduced the following concept of spherical \(t\)-designs. A finite subset \(X\) of \(S^{d-1}\) is called a spherical \(t\)-design, if \[ {1\over\sigma(S^{d-1})}\int_{S^{d-1}}f(x)d\sigma(x)={1\over| X|}\sum_{x\in X}f(x) \] holds for all polynomials \(f(x)=f(x_ 1,\dots,x_ d)\) of degree at most \(t\). Here \(\sigma\) denotes the surface measure on \(S^{d-1}\). The existence of spherical \(t\)-designs for every \(t\) and \(d\) was proved by \textit{P. D. Seymour} and \textit{T. Zaslawsky} [Adv. Math. 52, 213-240 (1984; Zbl 0596.05012)]. Since spherical designs can be used for numerical integration it is of interest to explicitly construct spherical \(t\)-designs \(X\) for all possible values of \(t\), \(d\), \(N=| X|\). However, only some sporadic examples of spherical \(t\)- designs are known, for specific values of \(t\), \(d\) and \(N\), and no general construction has been given. In this paper the author gives an explicit construction of spherical \(t\)-designs on \(S^{d-1}\) containing \(N\) points, for every \(t\), \(d\) and \(N\), \(N\geq N_ 0\), where \(N_ 0=C(d)^{d^ 3/2-d/2-3}\) for some function \(C(d)\).
scientific article; zbMATH DE number 93781