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Spherical quadrature formulas with equally spaced nodes on latitudinal circles.

AMS subject classifications. 65D32, 43A90, 42C10

1. Introduction. Let [S.sup.2] = {x [member of] [R.sup.3] : [parallel]x[[parallel].sub.2] = 1} denote the unit sphere of the Euclidean space [R.sup.3] and let

[psi] : [0,[pi]] x [0, 2[pi]) - [S.sup.2],

([rho],[theta]) [??] (sin [rho] cos [theta], sin [rho] sin [theta], cos [rho])

be its parametrization in spherical coordinates ([rho], [theta]). The coordinate [rho] of a point [xi]([PSI]([rho], [theta])) [member of] [S.sup.2] is usually called the latitude of [xi]. Let [P.sub.k], k = 0,1, ..., denote the Legendre polynomials of degree k on [-1,1] normalized by the condition [P.sub.k] (1) = 1, and let [V.sub.n] be the space of spherical polynomials of degree less than or equal to n. The dimension of [V.sub.n] is dim [V.sub.n] = [(n + 1).sup.2] and an orthogonal basis of [V.sub.n] is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Here [P.sup.D.sub.m] denotes the associated Legendre functions, defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For given functions f,g : [S.sup.2] [right arrow] C, the inner product is taken as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where dw([xi]) stands for the surface element of the sphere. We also denote by [[PI].sub.n] the set of univariate polynomials of degree less than or equal to n.

2. Spherical quadrature. Let n,p [member of] N, [[beta].sub.n] = ([[beta].sub.1], ..., [[beta].sub.n+1]) [member of] [[0,2[pi]).sup.n+1],

[[rho].sub.n] = ([[rho].sub.1], ..., [[rho].sub.n+1]), 0 < [[rho].sub.1] < [[rho].sub.2] < ... < [[rho].sub.n+1] < [pi], and let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

be a system of (p + 1) equally spaced nodes at each of the latitudes [[rho].sub.j]. We consider the quadrature formula,

(2.1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

with [[xi].sub.j,k] [member of] S ([[beta].sub.n], [[rho].sub.n], p).

A particular case, when n is odd, p = n, and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

with [alpha] [member of] [0,2),(see [1,2]) was already considered in [4]. Here the weights [w.sub.j] are uniquely determined and are calculated by direct manipulation of some Gram matrices of a local basis associated with the fundamental system of points S([[beta].sub.n], [[rho].sub.n], n). The quadrature formulas are interpolatory and therefore the degree of exactness is at least n. In [4] we showed that the degree of exactness is n +1 if and only if [alpha] = 1 and [[summation].sup.n+1.sub.j=1] [w.sub.j] [P.sub.n+1](cos [[rho].sub.j]) = 0. In [5] we proved that n + 1 is the maximal degree of exactness attained in this particular case.

In the following, for a fixed n, we wish to study the maximum degree of exactness which can be achieved with such a formula. This means to impose that (2.1) be exact for the spherical polynomials [Y.sup.l.sub.m], for l = -m, ..., m, and to specify the maximum value of m which makes (2.1) exact.

On the one hand, evaluating the integral in (2.1) for these spherical polynomials, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

On the other hand, evaluating the sum in (2.1) for these spherical polynomials, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The last sum is zero if l [not member of] (p + 1)Z and is p + 1 if l g (p + 1)Z.

With the above remarks, the quadrature formula (2.1) is exact for [Y.sup.l.sub.m] with l [not equal to] 0, in the case when m < p + 1. In order to be exact for l = 0 we should have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

With the notation cos [[rho].sub.j] = [r.sub.j], [a.sub.j] = p+1/2[pi] [w.sub.j], we arrive at

(2.2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In conclusion, we proved the following result.

Proposition 2.1. Let n,p, s [member of] N such that s < p + 1, and consider the spherical quadrature formula (2.1) with [[xi].sub.j,k] [member of] S ([[beta].sub.n], [[rho].sub.n], p). This formula is exact for the spherical polynomials in [V.sub.s] if and only if the quadrature formula

(2.3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is exact for all polynomials in [[PI].sub.s].

Let us remark that, taking m = 0,1, ..., p in (2.2) (or, equivalently, taking f = 1, x, ..., [x.sup.p] in (2.3)), we obtain the system

(2.4) [n+1.summation over (j=1)] [a.sub.j] [r.sup.[lambda].sub.j] = ((- 1).sup.[lambda]] + 1) 1/[lambda] + 1,

for [lambda] = 0, ..., p. This system has p + 1 equations and 2n + 2 unknowns, [a.sub.j], [r.sub.j], j = 1, ..., n + 1.

Next it is natural to ask when formula (2.1) is exact for spherical polynomials in [V.sub.s] with s [greater than or equal to] p +1. If we further impose that formula (2.1) is exact for the spherical polynomials [Y.sup.l.sub.p+1], l = -p - 1, ..., p + 1, then we have

(2.5) [n+1.summatio over j=1)] [a.sub.j] [r.sup.p+1.sub.j] = [((-1).sup.p+1] + 1) 1/p+2,

(2.6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Equation (2.5) follows from the fact that (2.1) is exact for [Y.sup.0.sub.p+1], while equation (2.6) results from the fact that formula (2.1) is exact for the spherical polynomials [Y.sup.p+1.sub.p+1] and [Y.sup.-p-1.sub.p+1]. For l = -p, ..., -1,1, ..., p, both sides of quadrature (2.1) are zero, therefore it is exact.

In conclusion the following proposition holds.

Proposition 2.2. Let n,p [member of] N. Then formula (2.1) is exact for all spherical polynomials in [V.sub.p] if and only if conditions (2.4) are satisfied for [lambda] = 0, ..., p. Moreover, formula (2.1) is exact for all spherical polynomials in [V.sub.p+1] if and only if supplementary conditions (2.5) and (2.6) are fulfilled.

3. Maximal degree of exactness which can be attained with equally spaced nodes at n + 1 latitudes. In this section we establish which is the maximum degree of exactness that can be obtained by taking the same number of equally spaced nodes on each of the n + 1 latitudinal circles and then we construct quadrature formulas with maximal degree of exactness.

What is well known is that the system (2.4) is solvable for a maximal number of conditions 2n + 2 (for [lambda] = 0,1,..., 2n +1), when it solves uniquely. This is the case of the univariate Gauss quadrature formula. In this case, the maximal value for p which can be taken in (2.4) is p = 2n +1, implying that (2.1) is exact for all spherical polynomials in [V.sub.2n+1]. In conclusion, the following result holds.

Proposition 3.1. Let n [member of] N and consider the quadrature formula (2.1). Its maximal degree of exactness is 2n + 1 and ifwe want it to be attained, then we must take the cosines of the latitudes, cos [[rho].sub.j] = [r.sub.j], as the roots of the Legendre polynomial [P.sub.n+1] and the weights as [3]

(3.1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

One possible case when it can be attained is by taking 2n + 2 equally spaced nodes at each latitude and arbitrary deviations [[beta].sub.j] [member of] [0, 2[pi]).

The question which naturally arises is whether we can obtain degree of exactness 2n +1 with fewer than 2n + 2 points at each latitude.

3.1. Maximal exactness 2n +1 with only 2n +1 nodes at each latitude. Consider 2n +1 equally spaced nodes at each latitude. If we suppose that conditions (2.4) are satisfied for [lambda] = 0,1, ..., 2n, then formula (2.1) will be exact for all spherical polynomial in [V.sub.2n]. From Proposition 2.2 we deduce that, if we want it to be exact for all polynomials in [V.sub.2n+i], then we should add the conditions

(3.2) [n+1.summation over (j=1)] [a.sub.j] [r.sup.2n+1.sub.j] = 0,

(3.3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In this case the quadrature formula (2.2) becomes the Gauss quadrature formula. Thus, [r.sub.j] will be the roots of the Legendre polynomial [P.sub.n+i] and [a.sub.j] are given in (3.1). Since [a.sub.n+2-j] = [a.sub.j] and [[rho].sub.j] = [pi] - [[rho].sub.n+2-j] for j = 1, ..., n + 1 and r n/2+1 = 0 for even n, condition (3.3) can be written as

(3.4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], 0, for n odd,

(3.5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] = 0, for n even.

For n odd, equation (3.4) is always solvable and possible solutions are discussed in Appendix A. For n even the solvability of equation (3.5) is discussed in Appendix B. Numerical tests performed for n [less than or equal to] 100 show that inequality (B.3) in Appendix B holds only for n [greater than or equal to] 12. Therefore, the equation (3.5) is not solvable for n [member of] {2,4, ..., 10} and solvable for 12 [less than or equal to] n [less than or equal to] 100. In conclusion, the following result holds.

Proposition 3.2. Let n [member of] N and consider the quadrature formula (2.1) with 2n +1 equally spaced nodes at each latitude. For n [member of] {2, 4, 6, 8,10} one cannot attain exactness 2n + 1. For n odd and for n [member of] {12, 14, ..., 100}, if cos [[rho].sub.j] are the roots of the Legendre polynomial [P.sub.n+i], the weights are as in (3.1), the numbers [[beta].sub.j] are solutions of equation (3.3) (given in Appendices 1 and 2), then the quadrature formula (2.1) has the degree of exactness 2n + 1.

We further want to know if it is possible to obtain the maximal degree of exactness 2n +1 with fewer points at each latitude.

3.2. Maximal exactness 2n +1 with 2n points at each latitude. Let us consider 2n points (p = 2n - 1) at each latitude. If we suppose that conditions (2.4) are satisfied for [lambda] = 0,1, ..., 2n - 1, then formula (2.1) will be exact for all polynomials in [V.sub.2n-i]. If we want it to be exact for [Y.sup.l.sub.2n], for l = -2n,..., 2n, then we should add the conditions

(3.6) [n+1.summation over (j=1)] [a.sub.j] [r.sup.2n.sub.j] = 2/2n + 1.

(3.7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Further, if we want the formula (2.1) to be exact for all [Y.sup.l.sub.2n+i], for l = -2n - l, ..., 2n + 1, then we should impose the conditions

(3.8) [n+1.summation over (j=1)] [a.sub.j] [r.sup.2n+1.sub.j] = 0,

(3.9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

From conditions (3.6) and (3.8) we get again that cos [[rho].sub.j] = [r.sub.j] are the roots of the Legendre polynomial [P.sub.n+i] and [a.sub.j] are as in (3.1). Therefore, formula (2.1) has the degree of exactness 2n +1 if and only if equations (3.7) and (3.9) are simultaneously satisfied. Due to the symmetry, they reduce to the system

(3.10) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

(3.11) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

for n odd, and to the system

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

for n even.

For n odd, we give some conditions on the solvability or non-solvability of this system in Appendix C (Proposition C. 1). Numerical tests performed for n [member of] {1, 3,5, ..., 99} show that the hypotheses (C.4) in Appendix C are fulfilled only for n [member of] {1,3, ..., 13}, in each of these cases the index k being k = (n + 1)/2. In conclusion, for these values of n, the above system has no solution and therefore the quadrature formula cannot have maximal exactness 2n + 1.

For n [member of] {15,17, ..., 41} the system is solvable since hypotheses (C.7)-(C.8) in Appendix C are fulfilled, each time for v = (n + 1)/2. In the proof of Proposition C.1, 3 in Appendix C, we give a possible solution of the system. For n {43, 45, . . . , 99}, the solvability is not clear yet. In this case, both sequences {[[alpha].sub.j], j = 1, ..., (n + 1)/2} and {[[mu].sub.j], j = 1,..., (n + 1)/2} satisfy the triangle inequality.

In Table 3.1 we summarize all the cases discussed above.

As a final remark, we mention that the improvement brought to the interpolatory quadrature formulas in [4], which were established only for n odd, is the following: In [4], for attaining the degree of exactness 2n +1 one needs [(2n + 2).sup.2] nodes. The quadrature formulas presented here can attain this degree of exactness with only (2n + 2)(n + 1) nodes (for arbitrary choices of the deviations [[beta].sub.j]) and with only (2n + 1)(n +1) nodes or only 2n(n + 1) nodes (for some special cases summarized in Table 3.1).

4. A particular case: spherical designs. A spherical design is a set of points of [S.sup.2] which generates a quadrature formula with equal weights which is exact for spherical polynomials up to a certain degree. For a fixed n [member of] N, we intend to specify the maximal degree of exactness that can be attained with the points in S([[beta].sub.n], [[rho].sub.n], p) and show for which choices of the parameters [[beta].sub.n], [[rho].sub.n], p this maximal degree can be attained. Therefore, let us consider the quadrature formula

(4.1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

If we require that this formula is exact for constant functions, we obtain

[w.sub.n,P] = 4[pi]/(n+1)(p+1)

As in the general case, we obtain that formula (4.1) is exact for the spherical polynomials [Y.sup.l.sub.m] for m < p +1 and -m [less than or equal to] l [less than or equal to] m, l [not equal to] 0. In order to be exact for [Y.sup.0.sub.m] for m < p + 1, we should have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [r.sub.j] = cos [[rho].sub.j], for j = , ..., n + . In conclusion, if the quadrature formula

(4.2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is exact for all univariate polynomials in [[PI].sub.s], s < p + 1, then the quadrature formula (4.1) will be exact for all spherical polynomials in [V.sub.s]. If in (4.2) we take f (x) = [x.sup.m] for m = 1, ..., p, we obtain the system

(4.3) [n+1.summation over (j=1)] [r.sup.[lambda].sub.j] = (- 1).sup.[lamnda]] + 1/[lambda] + 1 . n + 1/2,

with [lambda] = 1 ,..., p. This system has n + 1 unknowns. The maximal degree of exactness of the quadrature formula (4.2) (respectively, the maximal value ofp) is obtained in the classical case of Chebyshev one-dimensional quadrature formula, when the system (4.3) has a unique solution. In this case p = n + 1 , since the number of conditions needed to solve the quadrature formula uniquely is n + 1 . More precisely, in the one-dimensional case of Chebyshev quadrature, it is known that [r.sub.j] = [r.sub.n+2-j] for j = 1, ..., [n/2] and that system (4.3) has no solution for n = 7 and n > 8. For n [member of] {2,4, 6,8}, the quadrature formula (4.2) has the degree of exactness n +1 if the conditions in (4.3) are fulfilled for [lambda] = 1, ..., n + 1. For n [member of] {1, 3, 5}, if the same conditions are fulfilled, the degree of exactness is n + 2 since one additional condition in (4.3) for [lambda] = n + 2 is satisfied.

In conclusion, the following result holds.

Proposition 4.1. Let n [member of] {1, 2, 3, 4, 5, 6, 8} and consider the quadrature formula (4.1) with p + 1 equally spaced nodes at each latitude. Its maximal degree of exactness is

(4.4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

It can be attained, for example, by taking n + 2 equally spaced nodes at each latitude (p = n + 1), for all choices of the deviations [[beta].sub.j] in [0, 2[pi]) and for cos [[rho].sub.j] the nodes of the classical one-dimensional Chebyshev quadrature formula.

We wish to investigate if the maximal degree of exactness [[micro].sub.max] can be obtained with fewer than n + 2 points at each latitude

4.1. Maximal degree of exactness attained with only n + 1 points at each latitude.

Suppose p = n and suppose (4.3) is fulfilled for [lambda] = 1, ..., n. This implies that (4.1) is exact for the spherical polynomials [Y.sup.0.sub.[lambda]], for [lambda] = 1, ..., n. We want again to investigate if the maximal degree of exactness [[micro].sub.max] can be attained with only n + 1 points at each latitude.

Case 1: n even. If we want formula (4.1) to be exact for all spherical polynomials in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], it remains to impose the condition that (4.1) is exact for [Y.sup.0.sub.n+i] and [Y.sup.[+ or -](n+1).sub.n+1]. Exactness for [Y.sup.0.sub.n+1] means [[summation].sup.n+1.sub.j=1] [r.sup.n+1.sub.j] = 0, which, together with (4.3) fulfilled for [lambda] = [r.sub.j] = [r.sub.n+2-j], for j = 1, ..., n, 0 leads finally to the system in the classical one-dimensional Chebyshev case. Thus [r.sub.j] = [r.sub.n+2-j], for j = 1, ..., n/2, r n/2 + 1 = 0 and a solution exists only for n [member of] {2,4, 6,8}.

Further, exactness for [Y.sup.[+ or -](n+i).sub.n+1] reduces to

(4.5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Numerical tests show that condition (B.3) in Appendix B is fulfilled for n G {2,4, 6,8}. Therefore, equation (4.5) is solvable.

Case 2: n odd. In this case, if we want formula (4.1) to be exact for all spherical polynomials in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], it remains to require that it is exact for [Y.sup.0.sub.n+1], [Y.sup.0.sub.n+2], [Y.sup.[+ or -](n+i).sub.n+1] and [Y.sup.[+ or -](n+1).sub.n+2].

Exactness for the spherical polynomial [Y.sup.0.sub.n+1] reduces to the condition

[n+1.summation over (j=1)] [r.sup.n+1.sub.j] = n + 1/n + 2,

which, added to conditions (4.3) for [lambda] = 1, ..., n, leads again to the system in the classical one-dimensional Chebyshev case (which is uniquely solvable).

Exactness for [Y.sup.0.sub.n+2] reduces to condition

[n+1.summation over (j=1)] [r.sup.n+2.sub.j] = 0,

which is automatically satisfied.

Further, exactness for [Y.sup.[+ or -](n+1).sub.n+1] and [Y.sup.[+ or -](n+1).sub.n+2]) means, respectively,

(4.6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(4.7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In conclusion, the maximal degree of exactness n + 2 is attained if and only if r j are the nodes in univariate Chebyshev quadrature and the system (4.6)-(4.7) is solvable. The solvability of this system is discussed in Appendix C in the general case. For n = 1 , the non-solvability is clear. For n = 3, the system is again not solvable (cf. Proposition C.1, Appendix C), since [[mu].sub.1] < [[mu].sub.2]. For n = 5, it is solvable since the hypotheses (C.5)-(C.6) in Proposition C.1 are satisfied, with v = 2.

To summarize the above considerations, we state the following result.

Proposition 4.2. Let n [member of] {1, 2,3, 4,5, 6,8} and consider the quadrature formula (4.1) with n +1 equally spaced nodes at each latitude. Then the maximal degree of exactness Mmax given in Proposition 4.1 can be attained for n = 2, 4, 6, 8, if cos [[rho].sub.j] are chosen as nodes of the classical one-dimensional Chebyshev quadrature formula and the numbers [[beta].sub.j] are chosen as described in Appendix B. For n = 1, 3, the maximal degree of exactness cannot be attained, while for n = 5 it can be attained if the deviations [[beta].sub.j], j = 1 , ..., 6, are taken as described in Appendix C, Proposition C. 1, 2.

The natural question which arises now is: Is it possible to have maximal degree of exactness n + 1 with only n points at each latitude? The answer is given in the following section.

4.2. Maximal degree of exactness with only n points at each latitude. Let us consider n points at each latitude (p = n - 1) and suppose (4.3) holds for [lambda] = 1, ..., n - 1. We want to see if the maximal degree of exactness [[micro].sub.max] can be attained with only n points at each latitude.

Case 1: n odd. In this case, if we want formula (4.1) to be exact for all spherical polynomials in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], it remains to impose that it is exact for [Y.sup.0.sub.n+1], [Y.sup.0.sub.n+2], [Y.sup.[+ or -]n.sub.n+1] [Y.sup.[+ or -]n.sub.n+1] and [Y.sup.[+ or -]n.sub.n+2]. Altogether, they imply that [r.sub.j] = cos [[rho].sub.j] are the abscissa in the classical univariate Chebyshev case, and the deviations [[beta].sub.j] should satisfy the system

(4.8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

(4.9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since [P.sup.(n),sub.(+)2] (cos [rho]) is an even polynomial of degree two in cos [rho], using equation (4.8), we can replace the last equation by

(4.10) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For n = 1 , the system is clearly not solvable.

For n = 3, the system is solvable since [sin.sup.3] [[rho].sub.1] cos [[rho].sub.1] = [sin.sup.3] [[rho].sub.2] cos [[rho].sub.2]. A solution can be written as

[[beta].sub.1] [member of] [0, 2[pi]), [[beta].sub.3] = [[beta].sub.1], [[beta].sub.2] = [[beta].sub.4] = [[beta].sub.1] + [pi] (mod 2[pi]).

For n = 5, up to now we do not have a result regarding the solvability of the system.

Case 2: n even. If we want formula (4.1) to be exact for all spherical polynomials in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], it remains to impose that (4.1) is exact for [Y.sup.0.sub.n], [Y.sup.0.sub.n+1], [Y.sup.[+ or -]n.sub.n] and [Y.sup.[+ or -]n.sub.n+1]. Exactness for [Y.sup.0.sub.n] and [Y.sup.0.sub.n+1] means [[summation].sup.n+1.sub.j=1] = 1 and [[summation].sup.n+1.sub.j=1] [r.sup.n+1.sub.j] = 0, respectively. Together with (4.3) fulfilled for [lambda] = 1, ..., n - 1, they lead to the system in the classical one-dimensional Chebyshev case. Thus [r.sub.j] = [r.sub.n+2-j], for j = 1, ..., n/2, r n/2 + 1 = 0 and a solution exists only for n [member of] {2,4, 6,8}. Further, using again the symmetry of the latitudes, exactness for

[Y.sup.[+ or -]n.sub.n] and [Y.sup.[+ or -]n.sub.n+1] reduces to

(4.11) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

(4.12) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In conclusion, the maximal degree of exactness [[mu].sub.max] = n + 1 can be attained if and only if the system (4.11)-(4.12) is solvable. Unfortunately we could not give a result regarding the solvability of this system.

All these cases are summarized in Table 4.1.

5. Numerical examples. In order to demonstrate the efficiency of our formulas, we consider the quadrature formula

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

with [[xi].sub.j,k]([[rho].sub.j], [[theta].sup.j.sub.k]) [member of] [S.sup.2] in the following cases:

1. The classical Gauss-Legendre quadrature formula, with m = n, p = 2n + 1,

cos [[rho].sub.j] = [r.sub.j], the roots of Legendre polynomial [P.sub.n+1],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

j = 1, ..., n + 1, k = 1, ..., 2n +2. This formula has 2[n.sup.2] + 4n +2 nodes and is exact for polynomials in [V.sub.2n+1]. It is in fact a particular case of the quadratures given in Proposition 3.1, when all deviations [[beta].sub.j] are zero.

2. The Clenshaw-Curtis formula (1), with m = 2n, p = 2n + 1,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This formula has 4[n.sup.2] + 6n + 2 nodes and is exact for polynomials in [V.sub.2n+1].

In our numerical experiments we have considered the following test functions:

[f.sub.1](x) = -5sin(1 + [10x.sub.3]),

[f.sub.2] (x) = [parallel]x[[parallel].sub.1]/10,

[f.sub.3](x) = 1/[parallel]x[[parallel].sub.1],

[f.sub.4](x) = exp([x.sup.2.sub.1]),

where x = ([x.sub.1], [x.sub.2], [x.sub.3]) [S.sup.2].

From the quadrature formulas constructed in this paper, we consider those from Section 3.1 and we compare them with the Gauss-Legendre and Clenshaw-Curtis quadratures mentioned above. We do not present here quadratures from Proposition 3.1 for deviations [[beta].sub.j] different from zero, since in this case, for the above test functions, the errors are comparable with the ones obtained for Gauss-Legendre (when all [[beta].sub.j] are equal to zero).

Figure 5.1 shows the interpolation errors (logarithmic scale) for each of the functions [f.sub.1], [f.sub.2], [f.sub.3], and [f.sub.4], respectively.

Appendix A. For n odd, we provide solutions of the equation

(A.1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

with q = (n + 1)/2, [[alpha].sub.j] > 0 given and the unknowns [[beta].sub.j], j = 1, ..., n + 1. For this we need the following result.

Lemma A.1. Let A > 0 be given. Then, for every z = [te.sup.i[theta]] [member of] C with 0 [less than or equal to] t [less than or equal to] 2A, [theta] [member of] [0, 2[pi]), there exist [w.sub.j] = [w.sub.j] (t, [theta]) [member of] [0, 2[pi]), j = 1, 2, such that

(A.2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. Indeed, denoting

[gamma] = arccos [tau]/2A [member of] [0, [pi]/2],

a possible choice of the [w.sub.1], [w.sub.2] which satisfy relation (A.2) is the following:

1. If [theta] - [gamma] [greater than or equal to] 0 and [theta] + [gamma] < 2[pi], then ([w.sub.1], [w.sub.2]) [member of] {([theta] + [gamma], [theta] - [gamma]), ([theta] - [gamma], [theta] + [gamma])};

2. If [theta] - [gamma] < 0, then ([w.sub.1], [w.sub.2]) [member of] {([theta] + [gamma], [theta] - [gamma] + 2[pi]), ([theta] - [gamma] + 2[pi], [theta] + [gamma])};

3. If [theta] + [gamma] [greater than or equal to] 2[pi], then ([w.sub.1], [w.sub.2]) [member of] {([theta] + [gamma] - 2[pi], [theta] - [gamma]), ([theta] - [gamma], [theta] + [gamma] - 2[pi])}, or, shorter,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Equality (A.2) can be verified by direct calculations.

Let us come back to equation (A.1). For j = 1, ..., q, we consider [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [member of] C with 0 [less than or equal to] [t.sub.j] < 2[[alpha].sub.j], such that

[z.sub.1] + ... + [z.sub.q] = 0.

In fact, we take q - 1 arbitrary complex numbers [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and then consider [z.sup.*.sub.q] = -[z.sup.*.sub.1] - ... - [z.sup.*.sub.q-1]. The numbers [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], satisfying the inequalities [t.sub.j] < 2[[alpha].sub.j] are taken such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Denoting

[[gamma].sub.j] = arccos [t.sub.j]/2[[alpha].sub.j], j = 1, ..., q.

and applying Lemma A.1, we can write a solution of equation (A. 1) as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[FIGURE 5.1 OMITTED]

Appendix B. For n even, we discuss the equation

(B.1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

with q = n/2, [[alpha].sub.j] > 0 given and the unknowns [[beta].sub.j], j = 1 , . . . , q + 1 . For determining a non-trivial solution we need the following result.

LEMMA B. 1. Let a, [b.sub.1] ,..., [b.sub.q] > 0 such that a [less than or equal to] [b.sub.1] + ... + [b.sub.q]. Then there exist numbers [t.sub.j] G [0,1] (not all of them equal) for j [member of] {1, ..., q}, such that

(B.2) a = [q.summation over (j=1)] [t.sub.j] [b.sub.j].

Proof. Of course, a trivial solution, when all [t.sub.j] are equal, is

[t.sub.j] = [t.sup.*] = a/[b.sub.1] + ... + [b.sub.q] [member of] (0,1], for j = 1,2, ..., q + 1,

and it leads to a trivial solution of (3.5).

For non-trivial solutions, let t = a([b.sub.1] + ... + [[b.sub.q]).sup.-1] G (0,1]. There exist [[epsilon].sub.j] [member of] [0, t], j = 1, ..., q - 1 such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The numbers [t.sub.j], defined as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

satisfy the equality (B.2).

We will prove that equation (B. 1) is solvable if and only if

(B.3) [[alpha].sub.q+1] [less than or equal to] 2 [q.summation over (j=1)] [[alpha].sub.j].

Indeed, if the equation is solvable, (B.3) follows immediately by applying the triangle inequality. Conversely, suppose that (B.3) holds. From the previous lemma, there exist numbers [t.sub.j] [member of] [0,1] such that [[alpha].sub.q+1] = 2 [[summation].sup.q.sub.j=1] [[alpha].sub.j] [t.sub.j]. Then a solution of equation (B.1) is

[[beta].sub.j] = arccos [t.sub.j], [[beta].sub.n+2-j] = 2[pi] - [[beta].sub.j] (mod 2[pi]), for j = 1, ..., q,

[[beta].sub.q+1] = [pi].

Appendix C. For n odd, we discuss the solutions of the system

(C.1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

(C.2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

with q = n+1/2, [[alpha].sub.j], [[mu].sub.j] > 0 given and [x.sub.j], [y.sub.j] [member of] [0, 2[pi]) unknowns. Due to our particular problems (systems (3.10)-(3.11) and (4.6)-(4.7)), we will also suppose that

(C.3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For n = 1 the incompatibility is immediate, so let us suppose in the sequel that n [greater than or equal to] 3.

Proposition C.1. Under the above assumptions, the following statements are true:

1. If there exists k [member of] {1 , ..., q} such that

(C.4) [[alpha].sub.k] [[mu].sub.k] > [[alpha].sub.k] [k-1.summation over (j=1)] [[mu].sub.j] + [[mu].sub.k] [q.summation over (j=k+1)] [[alpha].sub.j],

then the system (C.1)-(C.2) is not solvable.

2. If there exists v [member of] {1 ,..., q} such that

(C.5) [[mu].sub.v] [greater than or equal to] [q.summation over (j=1,j[not equal to]v)] [[mu].sub.j],

(C.6) [[alpha].sub.v] [less than or equal to] [q.summation over (j=1,j[not equal to]v)] [[alpha].sub.j]

then the system is solvable.

3. If there exists v [member of] {1, ..., q} such that

(C.7) [[alpha].sub.v] [greater than or equal to] [q.summation over (j=1,j[not equal to]v)] [[alpha].sub.j],

(C.8) [[mu].sub.v] [less than or equal to] [q.summation over (j=1,j[not equal to]v)] [[mu].sub.j],

then the system is solvable.

Proof.

1. We suppose that the system is solvable and let [x.sub.j], [y.sub.j], j = 1,..., q, be a solution. If we multiply the equations (C.1)-(C.2) by [[mu].sub.k] and [[alpha]].sub.k], respectively, and then we add them, we get, for all k = 1 , ..., q,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Using the triangle inequality and the identity a + b + [absolute value of a - b] = 2 max{a, b}, we obtain

[[alpha].sub.k] [[mu].sub.k] [less than or equal to] [q.summation over (j=1,j[not equal to]k)] max{[[alpha].sub.k] [[mu].sub.k], [[alpha].sub.j] [[mu].sub.k]}.

Using now the hypothesis (C.3), this inequality can be written as

[[alpha].sub.k] [[mu].sub.k] [less than or equal to] [[alpha].sub.k] [k-1.summation over (j=1)] [[mu].sub.j] + [[mu].sub.k] [q.summation over (j=k+1)] [[alpha].sub.j],

which contradicts (C.4). In conclusion, the system is incompatible.

2. Applying Lemma B.1, there are numbers [t.sub.j] [not member of] [0,1], j = 1, ..., q, j not equal to] v, such that

[[alpha].sub.v] = [summation over (j=1,j[not equal to]v] [[alpha].sub.j] [t.sub.j].

We define the function [phi]: [0, 2] [right arrow] R,

[phi](t) = [q.summation over (j=1,j[not equal to]v)] [[mu].sub.j] [square root of 4 - [t.sup.2.sub.j][t.sup.2]] - [[mu].sub.v] [square root of 4 - [t.sup.2]].

Since [phi]>(0) x [phi](2) [less than or equal to] 0, there exists [t.sub.0] [member of] [0,2] such that [phi]([t.sub.0]) = 0. A simple calculation shows that a solution of the system can be written as

[x.sub.j] = arccos [t.sub.0][t.sub.j]/2, [y.sub.j] = 2[pi] - [x.sub.j](mod2[pi]), for j [not equal to] v.

[x.sub.v] = [pi] + arccos [t.sub.0]/2, [y.sub.v] = [pi] - arccos [t.sub.0]/2.

3. Let [t.sub.1] = [[alpha].sup.-1.sub.v] [[summation].sup.q.sub.j=1,j[not equal to]v] [[alpha].sub.j] [less than or equal to] 1 and define the function [phi] : [0,1] [right arrow] R,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since [phi](0) x [phi](1) [less than or equal to] 0, there exists [t.sub.0] [member of] [0,1] such that [phi]([t.sub.0]) = 0. Then we define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

A simple calculation shows that a solution of the system can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Acknowledgment. This work was supported by a research scholarship from DAAD (German Academic Exchange Service), at the Institute of Mathematics in Lubeck. I thank Jurgen Prestin for his support during this scholarship.

REFERENCES

[1] N. LAIN FERNANDEZ, Polynomial Bases on the Sphere, Logos-Verlag, Berlin, 2003.

[2] --, Localized polynomial bases on the sphere, Electron. Trans. Numer. Anal., 19 (2005), pp. 84-93. http://etna.math.kent.edu/vol.19.2 005/pp84-93.dir

[3] G. Szego, Orthogonal Polynomials, Colloquium Publications, Vol. 23, American Mathematical Society, Rhode Island, 1975.

[4] J. Prestin and D. Rosca, On some cubature formulas on the sphere, J. Approx. Theory, 142 (2006), pp. 1-19.

[5] D. Rosca, On the degree of exactness of some positive cubature formulas on the sphere, Automat. Comput. Appl. Math., 15 (2006), pp. 279-283.

(1) This formula is sometimes called Chebyshev formula, since in the one-dimensional case it is based on the expansion of a function in terms of Chebyshev polynomials [T.sub.i] of the first kind. The nodes cos j[pi]/2n are the extrema of the Chebyshev polynomial [T.sub.2n] of degree 2n.

DANIELA ROSCA ([dagger])

* Received January 5, 2007. Accepted for publication May 18, 2009. Published online August 14, 2009. Recommended by S. Ehrich. This work was supported by DAAD scholarship at University of Lubeck, Germany.

([dagger]) Dept. of Mathematics, Technical University of Cluj-Napoca, str. Daicoviciu nr. 15, RO-400020 Cluj-Napoca, Romania (Daniela.Rosca@math.utcluj.ro).
Table 3.1
Some choices for which the maximal degree of exactness 2n + 1 is
attained, for [P.sub.n+1](cos [[rho].sub.j]) = 0, j [member of] {1,
..., n + 1}, n [less than or equal to] 100.

number of nodes           n                  [[beta].sub.j]
at each latitude

     2n+ 2                N                    [0, 2[pi])

     2n+ 1               odd                   Appendix A
                     {2,4,6,8,10}     [empty set] (cf. Appendix B)
                   {12,14, ..., 100}           Appendix B

       2n           {1,3, ..., 13}    [empty set] (cf. Appendix C,
                   {15,17, ..., 41}           Prop. C.l, 1)
                   {43,45, ..., 99}     Appendix C, Prop. C.l, 3
                         even                   no answer
                                                no answer

Table 4.1
Some choices for which the maximal degree of exactness
[[micro].sub.max] is attained, for cos [[rho].sub.j], j [member of]
{1, ..., n + 1}, the nodes in the case ofclassical Chebyshev
quadrature.

number of nodes           n                   [[beta].sub.j]
at each latitude

     n + 2         {1,2,3,4,5,6,8}               [0,2[pi])

     n + 1            {2,4,6,8}                  [0,2[pi])
                        {1,3}        0 (cf. Appendix C, Prop. C.1, 2)
                          5                      no answer

       n                  1                          0
                          3             [[beta].sub.1] [member of]
                      {2,4,6,8}         [0,2[pi]), [[beta].sub.3] =
                                     [[beta].sub.1], [[beta].sub.2] =
                                     [[beta].sub.4] = [[beta].sub.1] +
                                                   [pi]
                                                 no answer
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Author:Rosca, Daniela
Publication:Electronic Transactions on Numerical Analysis
Article Type:Report
Geographic Code:4EUGE
Date:Jan 1, 2009
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