Abstract
In 2000, Reimer proved that a positive weight quadrature rule on the unit sphere \(\mathbb{S}^{d} \subset \mathbb{R}^{{d + 1}} \) has the property of quadrature regularity. Hesse and Sloan used a related property, called Property (R) in their work on estimates of quadrature error on \(\mathbb{S}^{d}\). The constants related to Property (R) for a sequence of positive weight quadrature rules on \(\mathbb{S}^{d}\) can be estimated by using a variation on Reimer’s bounds on the sum of the quadrature weight within a spherical cap, with Jacobi polynomials of the form \(P^{{({1 + d} \mathord{\left/ {\vphantom {{1 + d} 2}} \right. \kern-\nulldelimiterspace} 2,d \mathord{\left/ {\vphantom {d 2}} \right. \kern-\nulldelimiterspace} 2)}}_{t} \), in combination with the Sturm comparison theorem. A recent conjecture on monotonicities of Jacobi polynomials would, if true, provide improved estimates for these constants.
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References
Andrews, G.E., Askey, R., Roy, R.: Special functions. In: Encyclopedia of Mathematics and Its Applications, vol. 71. Cambridge University Press, Cambridge (2000)
Brauchart, J.S., Hesse, K.: Numerical integration over spheres of arbitrary dimension. Constr. Approx. 25(1), 41–71 (2007)
Gautschi, W., Leopardi, P.: Conjectured inequalities for Jacobi polynomials and their largest zeros. In: DWCAA06 Proceedings in Numerical Algorithms (2007) doi:10.1007/s11075-007-9067-5
Hesse, K., Sloan, I.H.: Worst-case errors in a Sobolev space setting for cubature over the sphere S 2. Bull. Aust. Math. Soc. 71(1), 81–105 (2005)
Hesse, K., Sloan, I.H.: Cubature over the sphere S 2 in Sobolev spaces of arbitrary order. J. Approx. Theory 141(2), 118–133 (2006)
Le Gia, Q.T., Sloan, I.H.: The uniform norm of hyperinterpolation on the unit sphere in an arbitrary number of dimensions. Constr. Approx. 17, 249–265 (2001)
Makai, E.: On a monotonic property of certain Sturm–Liouville functions. Acta Math. Acad. Sci. Hung. 3, 165–172 (1952)
Müller, C.: Spherical harmonics. In: Lecture Notes in Mathematics, vol. 17, Springer, Berlin (1966)
Reimer, M.: Hyperinterpolation on the sphere at the minimal projection order. J. Approx. Theory 104, 272–286 (2000)
Reimer, M.: Multivariate polynomial approximation. In: International Series of Numerical Mathematics, vol. 144. Birkhäuser Verlag, Basel (2003)
Sloan, I.H.: Polynomial interpolation and hyperinterpolation over general regions. J. Approx. Theory 83, 238–254 (1995)
Sloan, I.H., Womersley, R. S.: Constructive polynomial approximation on the sphere. J. Approx. Theory 103, 91–118 (2000)
Szegö, G.: Orthogonal polynomials, 4th ed. In: American Mathematical Society Colloquium Publications, vol. 23. American Mathematical Society, Providence, RI (1975)
Watson, G.N.: A treatise on the theory of Bessel functions. Cambridge University Press, Cambridge, England (1944)
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The work was carried out while the author was a PhD student at the School of Mathematics, University of New South Wales.
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Leopardi, P.C. Positive weight quadrature on the sphere and monotonicities of Jacobi polynomials. Numer Algor 45, 75–87 (2007). https://doi.org/10.1007/s11075-007-9073-7
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DOI: https://doi.org/10.1007/s11075-007-9073-7