Quadrature formulas for integration of multivariate trigonometric polynomials on spherical triangles
Tóm tắt
We describe an explicit construction of quadrature rules exact for integrating multivariate trigonometric polynomials of a given coordinatewise degree on a spherical triangle. The theory is presented in the more general setting of quadrature formulas on a compact subset of the unit hypersphere,
$${\mathbb {S}^q}$$
, embedded in the Euclidean space
$${\mathbb {R} ^{q+1}}$$
. The number of points at which the polynomials are sampled is commensurate with the dimension of the polynomial space.
Tài liệu tham khảo
Freeden, W., Glockner, O., Schreiner M.: Spherical Panel Clustering and Its Numerical Aspects. AGTM Report No. 183, University of Kaiserlautern, Geomathematics Group (1997)
Freeden, W., Windheuser U.: Spherical Wavelet Transform and Its Discretization, AGTM Report No. 125, University of Kaiserlautern, Geomathematics Group (1995)
Freud, G.: Orthogonal Polynomials. Académiai Kiado, Budapest (1971)
Ganesh M., Mhaskar H.N.: Matrix-free interpolation on the sphere. SIAM J. Numer. Anal. 44(3), 1314–1331 (2006)
Gautschi W.: On generating orthogonal polynomials. SIAM J. Sci. Stat. Comput. 3(3), 289–317 (1982)
Gautschi W.: Orthogonal polynomials: computation and approximation. Oxford University Press, Oxford (2004)
Gautschi W.: Orthogonal polynomials (in Matlab). J. Comput. Appl. Math. 178(1–2), 215–234 (2005)
Hesse, K., Womersley, R.S.: Numerical integration with polynomial exactness over a spherical cap. Adv. Comput. Math. (2011) (available online Sept. 23, 2011)
Le Gia Q.T., Mhaskar H.N.: Localized linear polynomial operators and quadrature formulas on the sphere. SIAM J. Numer. Anal. 47(1), 440–466 (2008)
Mhaskar H.N., Narcowich F.J., Ward J.D.: Spherical Marcinkiewicz–Zygmund inequalities and positive quadrature. Math. Comput. 70, 1113–1130 (2001)
Mhaskar H.N.: Local quadrature formulas on the sphere. J. Complex. 20, 753–772 (2004a)
Mhaskar, H.N.: Local quadrature formulas on the sphere, II. In: Neamtu, M., Saff, E.B. (eds.) Advances in Constructive Approximation. Nashboro Press, Nashville, pp. 333–344 (2004b)
Renka R.J.: Algorithm 772: STRIPACK: Delaunay triangulation and Voronoi diagram on the surface of a sphere. ACM Trans. Math. Softw. 23(3), 416–434 (1997)