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Spherical quadrature formulas with equally-spaced nodes on latitudinal circles


  Daniela Rosca
    Department of Mathematics, Technical University of Cluj-Napoca
   ROMANIA

  given at  strobl07 (18.06.07 11:15)
  id:  755
  length:  25min
  status:  accepted
  type:  talk
  LINK-Presentation: 
  ABSTRACT:
In [2] we have constructed quadrature formulas on the 2-sphere, based on some fundamental systems of $(n+1)^2$ points ($n+1$ equally-spaced points taken on $n+1$ latitudinal circles), constructed by La\'{i}n-Fern\'{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 equally-spaced 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,q-1 $) and $x_j,y_j \in [0,2 \pi)$
unknowns. We give some sufficient conditions for its solvability, respectively non-solvability, 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.~84--93.

[2] {\sc J. Prestin and D. Ro\c{s}ca}, {\em On some cubature formulas on the
sphere}, J. Approx. Theory. 142 (2006), pp.~1--19.

[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.~283--288.


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