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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 \(\mu(t)>0\), \(t>0\).

MSC:
41A55 Approximate quadratures
41A63 Multidimensional problems
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[1] Aronszajn, N.; Greese, T.; Lipkin, L., Polyharmonic functions, (1983), Clarendon Press Oxford
[2] Bojanov, B., An extension of the Pizzetti formula for polyharmonic functions, Acta math. hungar., 91, 99-113, (2001) · Zbl 0980.31004
[3] B. Bojanov, Cubature formulae for polyharmonic functions, Recent Progress in Multivariate Approximation, (Witten Bommerholz, 2000), International Series of Numerical Mathematics, Vol. 137, Birkhäuser, Basel, 2001, pp. 49-74. · Zbl 0988.41017
[4] Bojanov, B.; Dimitrov, D., Gaussian extended cubature formulae for polyharmonic functions, Math. comp., 70, 671-683, (2000) · Zbl 0965.31007
[5] Bojanov, B.; Petrov, P., Gaussian interval quadrature formula, Numer. math., 87, 625-643, (2001) · Zbl 0969.41018
[6] Bojanov, B.; Petrova, G., Uniqueness of the Gaussian quadrature for a ball, J. approx. theory, 104, 21-44, (2000) · Zbl 0979.41020
[7] Cools, R., Ph. Rabinowitz, monomial cubature rules Since “stroud”a compilation, J. comput. appl. math., 48, 309-326, (1993) · Zbl 0799.65027
[8] Dimitrov, D., Integration of polyharmonic functions, Math. comp., 65, 1269-1281, (1996) · Zbl 0860.31003
[9] Dubiner, M., The theory of multi-dimensional polynomial approximation, J. anal. math., 67, 39-116, (1995) · Zbl 0857.41006
[10] M. Ganzburg, Polynomial approximation on the m-dimensional ball, Approximation Theory IX, Innov. Applied Mathematics, Vol. I, Vanderbilt Univ. Press, Nashville, TN 1998 pp. 141-148. · Zbl 0932.41005
[11] Nicolescu, M., LES fonctions polyharmoniques, (1936), Hermann Paris · JFM 62.0564.01
[12] Omladic, M.; Pahor, S.; Suhadolc, A., On a new type quadrature formulas, Numer. math., 25, 4, 421-426, (1976) · Zbl 0314.65007
[13] Petrova, G., Cubature formulae for the sphere and the ball in \(R\^{}\{n\}\), (), 380-384 · Zbl 1036.41017
[14] G. Petrova, Uniqueness of the Gaussian extended cubature for polyharmonic functions, East. J. Approx., to appear. · Zbl 1111.41023
[15] Pizzetti, P., Sulla media dei valore che una funzione dei punti Della spazio assume allasuperticie di una sfera, Rend. lincei, ser., V, XVIII, 1 sem, 182-185, (1909) · JFM 40.0455.01
[16] M. Reimer, On the existence-problem for Gauss-quadrature on the sphere, Approximation by Solutions of Partial Differential Equations (Hanstholm, 1991), NATO Advanced Science Institutes Series C Mathematical and Physical Sciences, Vol. 365, Kluwer Academic Publishers, Dordrecht, 1992, pp. 169-184. · Zbl 0759.41030
[17] Stroud, A., Approximate calculation of multiple integrals, (1971), Prentice-Hall Englewood Cliffs, NJ · Zbl 0379.65013
[18] Xu, Y., Orthogonal polynomials and cubature formulae on spheres and on simplices, Methods appl. anal., 5, 2, 169-184, (1998) · Zbl 0913.33005
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