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Cubature formulae for spheres, simplices and balls. (English) Zbl 1046.41015
The paper is organized in 5 Sections. The first two contain respectively the motivation for the choice of the subject and some results concerning properties of polyharmonic functions. In Section 3, cubature on the sphere \(S(r)=\{x\in\mathbb{R}^n:| x| =r\}\) on the form \[ \int_{S(r)}\mu(\xi) \,d\sigma(\xi)\simeq \sum^{m-1}_{j=0} A_j(r)\int_{S(r_j)}\mu(\xi) \,d\sigma(\xi), r\neq r_j \] for any distinct radii \(\{r,r_j\}^{m-1}_{j=0}\) are investigated. The uniqueness of the above formula which is exact for all \(u\in H^m(B(R))\), \(R:=\max_{j=0,\dots,m-1}\{r,r\}\) is proved and its weights \(A_j(r)\) are obtained. Section 4 is devoted to study the cubature \[ \int_{B(r)}\mu(| x| )u(x) \,dx\simeq\sum^{m-j}_{j=0} C_j(r) \int_{B(r_j)}\mu(| x| ) u(x) \,dx \] where \(B(r)=\{x\in\mathbb{R}^n:| x| =(\sum^n_{i=1} x_j^2)^{1/2}<r\}\) and the weight \(\mu\) is strictly positive \(\mu(t)>0\), \(t>0\).

41A55 Approximate quadratures
41A63 Multidimensional problems
Full Text: DOI
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