Информационная система "Конференции"

Abstracts

Approximation of functions and quadrature formulas

Cubature formulae begin{equation} frac{1}{4pi^2R}intlimits_Tf(x, y, z) dSsimeqsum_{i=1}^N C_i f(x_i,y_i,z_i) end{equation} are constructed for the torus $T$ in $R^3$ begin{equation} (x^2 + y^2 + z^2-R^2-1)^2 + 4R^2z^2-4R^2 = 0 end{equation} 4, 6 degree of accuracy invariant concerning the group $G$, generated by the reflections. The group $G=G_1times G_2$ is considered, where $G_1$ -- is generated by reflections from a plane $Oxy,$ and $G_2$ -- the group of conversions of a regular triangle in itself. For 4 degrees cubature formulae contains 18 nodes, for 6 - 36 nodes. For example, at $R = 1$ the formula of 4 degrees has the following coefficients and nodes. begin{center} begin{tabular}{|c|c||c|c|} hline Nodes & Coefficients & Nodes & Coefficients hline $(0,2,0)$ & $frac{23}{1152}$ & $(sqrt{3},1,0)$ & $frac{35}{384}$ hline $(sqrt{3},-1,0)$ & $frac{23}{1152}$ & $(-sqrt{3},1,0)$ & $frac{35}{384}$ hline $(-sqrt{3}, -1,0)$ & $ frac{23}{1152}$ & $(0,-2,0)$ & $frac{35}{384}$ hline $left(0,frac{3}{2},pmfrac{sqrt{3}}{2}right)$ & $frac{1}{12}$ & $left(0,frac{1}{2},pmfrac{sqrt{3}}{2}right)$ & $frac{1}{36}$ hline $left(frac{3sqrt{3}}{4},-frac{3}{4},pmfrac{sqrt{3}}{2}right)$ & $frac{1}{12}$ & $left(frac{sqrt3}{4},-frac{1}{4}, pmfrac{sqrt3}{2}right)$ & $frac{1}{36}$ hline $left(-frac{3sqrt3}{4}, -frac{3}{4}, pmfrac{sqrt{3}}{2}right)$ & $frac{1}{12}$ & $left(-frac{sqrt3}{4},-frac{1}{4},pmfrac{sqrt3}{2}right)$ & $frac{1}{36}$ hline end{tabular} end{center} begin{thebibliography}{1} bibitem{1} Misovskih I.P. Interpolational cubature formulae. M. Science, 1981.

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