Rotationally invariant quadratures for the sphere

We construct near-optimal quadratures for the sphere that are invariant under the icosahedral rotation group. These quadratures integrate all (N+1)2 linearly independent functions in a rotationally invariant subspace of maximal order and degree N. The nodes of these quadratures are nearly uniformly distributed, and the number of nodes is only marginally more than the optimal (N+1)2/3 nodes. Using these quadratures, we discretize the reproducing kernel on a rotationally invariant subspace to construct an analogue of Lagrange interpolation on the sphere. This representation uses function values at the quadrature nodes. In addition, the representation yields an expansion that uses a single function centred and mostly concentrated at nodes of the quadrature, thus providing a much better localization than spherical harmonic expansions. We show that this representation may be localized even further. We also describe two algorithms of complexity for using these grids and representations. Finally, we note that our approach is also applicable to other discrete rotation groups.

[1]  Jürgen Prestin,et al.  Interpolatory Band-Limited Wavelet Bases on the Sphere , 2005 .

[2]  Fr'ed'eric Guilloux,et al.  Practical wavelet design on the sphere , 2007, 0706.2598.

[3]  Catherine Constable,et al.  Foundations of geomagnetism , 1996 .

[4]  I. S. Gradshteyn,et al.  Table of Integrals, Series, and Products , 1976 .

[5]  Ronald Cools,et al.  Constructing cubature formulae: the science behind the art , 1997, Acta Numerica.

[6]  A. Stroud.  Approximate calculation of multiple integrals , 1973 .

[7]  S. L. Sobolev.  Cubature Formulas on the Sphere Invariant under Finite Groups of Rotations , 2006 .

[8]  V. I. Lebedev,et al.  Quadrature formula for the sphere of 131-th algebraic order of accuracy , 1999 .

[9]  Matthias Holschneider.  Poisson wavelets on the sphere , 2007, SPIE Optical Engineering + Applications.

[10]  I. Daubechies,et al.  Wavelets on irregular point sets , 1999, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[11]  C. T. Benson,et al.  Finite Reflection Groups , 1985 .

[12]  P. Swarztrauber,et al.  A standard test set for numerical approximations to the shallow water equations in spherical geometry , 1992 .

[13]  L. Greengard,et al.  The type 3 nonuniform FFT and its applications June - , 2005 .

[14]  Vladimir Rokhlin,et al.  Fast Fourier Transforms for Nonequispaced Data , 1993, SIAM J. Sci. Comput..

[15]  N. J. A. Sloane,et al.  Sphere Packings, Lattices and Groups , 1987, Grundlehren der mathematischen Wissenschaften.

[16]  Ian H. Sloan,et al.  Polynomial interpolation and hyperinterpolation over general regions , 1995 .

[17]  L. P. Pellinen.  Physical Geodesy , 1972 .

[18]  A. D. McLaren,et al.  Optimal numerical integration on a sphere , 1963 .

[19]  C. J. Bradley,et al.  On the symmetries of spherical harmonics , 1963, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[20]  Cubature formulas on a sphere invariant under the icosahedral rotation group , 2008 .

[21]  Quadratures of Gaussian type for a sphere invariant under the icosahedral group with inversion , 1979 .

[22]  Eugene Fiume,et al.  SOHO: Orthogonal and symmetric Haar wavelets on the sphere , 2008, TOGS.

[23]  Matthias Holschneider,et al.  Continuous wavelet transforms on the sphere , 1996 .

[24]  Sergei Sobolev.  Cubature formulas and modern analysis , 1993 .

[25]  M. Taylor.  The Spectral Element Method for the Shallow Water Equations on the Sphere , 1997 .

[26]  Ian H. Sloan,et al.  Constructive Polynomial Approximation on the Sphere , 2000 .

[27]  G. Beylkin.  On the Fast Fourier Transform of Functions with Singularities , 1995 .

[28]  Laurent Jacques,et al.  Wavelets on the sphere: implementation and approximations , 2002 .

[29]  J. Light,et al.  Generalized discrete variable approximation in quantum mechanics , 1985 .

[30]  Peter Schröder,et al.  Spherical wavelets: efficiently representing functions on the sphere , 1995, SIGGRAPH.

[31]  L. Greengard,et al.  Short Note: The type 3 nonuniform FFT and its applications , 2005 .

[32]  Lebedev discrete variable representation , 2007 .

[33]  R. Kress,et al.  Inverse Acoustic and Electromagnetic Scattering Theory , 1992 .

[34]  I. Sloan,et al.  Good approximation on the sphere, with application to geodesy and the scattering of sound , 2002 .

[35]  Bradley K. Alpert,et al.  A Fast Spherical Filter with Uniform Resolution , 1997 .

[36]  Gregory Beylkin,et al.  Toward Multiresolution Estimation and Efficient Representation of Gravitational Fields , 2002 .

[37]  Jack B. Kuipers,et al.  Quaternions and Rotation Sequences: A Primer with Applications to Orbits, Aerospace and Virtual Reality , 2002 .

[38]  P. Swarztrauber,et al.  A performance comparison of associated Legendre projections , 2001 .

[39]  Quoc Thong Le Gia,et al.  Localized Linear Polynomial Operators and Quadrature Formulas on the Sphere , 2008, SIAM J. Numer. Anal..

[40]  Willi Freeden,et al.  Constructive Approximation on the Sphere: With Applications to Geomathematics , 1998 .

[41]  V. I. Lebedev,et al.  Quadratures on a sphere , 1976 .