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Spherical quadrature formulas with equallyspaced nodes on latitudinal circles Daniela Rosca Department of Mathematics, Technical University of ClujNapoca ROMANIA given at strobl07 (18.06.07 11:15) id: 755 length: 25min status: accepted type: talk LINKPresentation: ABSTRACT:
In [2] we have constructed quadrature formulas on the 2sphere, based on some fundamental systems of $(n+1)^2$ points ($n+1$ equallyspaced points taken on $n+1$ latitudinal circles), constructed by La\'{i}nFern\'{a}ndez [1]. These quadratures are of interpolatory type, therefore the degree of
exactness is at least $n$. In some particular cases, the degree of exactness can be $n+1$ and this exactness is the maximal one which can be obtained, based on the above mentioned fundamental system of points [3].
In this paper we try to improve the exactness by taking more equallyspaced points on each latitude and equal weights for each latitude. We show that the maximal degree of exactness which can be attained with $n+1$ latitudes is $2n+1$, and then we present some situations in which this exactness can be achieved.
Of a special relevance is the discussion of solvability of the system
$$
\sum_{j=1}^q \alpha_j(e^{ix_j}+e^{iy_j})=0,\quad \sum_{j=1}^q
\mu_j(e^{ix_j}e^{iy_j})=0,
$$
with $n$ odd, $q= \frac{n+1}2,$ $\alpha _j,\mu_j>0$ satisfying the
inequalities $ \frac{\alpha_{j+1}}{\mu_{j+1}} \geq \frac
{\alpha_j}{\mu_j}$ ($j=1,\ldots,q1 $) and $x_j,y_j \in [0,2 \pi)$
unknowns. We give some sufficient conditions for its solvability, respectively nonsolvability, but we do not have yet a necessary and sufficient condition for its solvability.
[1] {\sc N. La\'{i}n Fern\'{a}ndez},
{\em Localized Polynomial Bases on the Sphere,}
Electron. Trans. Numer. Anal., 19 (2005), pp.~8493.
[2] {\sc J. Prestin and D. Ro\c{s}ca}, {\em On some cubature formulas on the
sphere}, J. Approx. Theory. 142 (2006), pp.~119.
[3] {\sc D. Ro\c{s}ca}, {\em On the degree of exactness of some positive cubature formulas on the sphere}, Aut. Comp. Appl. Math., 15 (2006), no. 1, pp.~283288.
