Fast memory efficient evaluation of spherical polynomials at scattered points

A method for fast evaluation of band-limited functions (spherical polynomials) at many scattered points on the unit 2-d sphere is presented. The method relies on the superb localization of the father needlet kernels and their compatibility with spherical harmonics. It is fast, local, memory efficient, numerically stable and with guaranteed (prescribed) accuracy. The speed is independent of the band limit and depends logarithmically on the prescribed accuracy. The method can be also applied for approximation on the sphere, verification of spherical polynomials and for fast generation of surfaces in computer-aided geometric design. It naturally extends to higher dimensions.

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

[2]  A. Erdélyi,et al.  Higher Transcendental Functions , 1954 .

[3]  Tamás Erdélyi,et al.  Generalized Jacobi weights, Christoffel functions, and Jacobi polynomials , 1994 .

[4]  Vilmos Totik,et al.  Fast decreasing polynomials , 1990 .

[5]  Martin J. Mohlenkamp.  A fast transform for spherical harmonics , 1997 .

[6]  Mark A Ratner,et al.  A fast method for solving both the time-dependent Schrödinger equation in angular coordinates and its associated "m-mixing" problem. , 2009, The Journal of chemical physics.

[7]  Witold Maciejewski,et al.  A MAGELLAN–IMACS-IFU SEARCH FOR DYNAMICAL DRIVERS OF NUCLEAR ACTIVITY. I. REDUCTION PIPELINE AND GALAXY CATALOG , 2011, 1112.4497.

[8]  Mark Tygert,et al.  Fast Algorithms for Spherical Harmonic Expansions , 2006, SIAM J. Sci. Comput..

[10]  Yuan Xu,et al.  Sub-exponentially localized kernels and frames induced by orthogonal expansions , 2008, 0809.3421.

[11]  Stefan Kunis,et al.  Fast spherical Fourier algorithms , 2003 .

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

[13]  Larry L. Schumaker,et al.  Spline functions on triangulations , 2007, Encyclopedia of mathematics and its applications.

[14]  Zydrunas Gimbutas,et al.  A fast and stable method for rotating spherical harmonic expansions , 2009, J. Comput. Phys..

[15]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

[16]  Vladimir Rokhlin,et al.  A Fast Algorithm for the Calculation of the Roots of Special Functions , 2007, SIAM J. Sci. Comput..

[17]  Mark Tygert,et al.  Fast algorithms for spherical harmonic expansions, II , 2008, J. Comput. Phys..

[18]  D. Healy,et al.  Computing Fourier Transforms and Convolutions on the 2-Sphere , 1994 .

[19]  E. LESTER SMITH,et al.  AND OTHERS , 2005 .

[20]  George G. Lorentz,et al.  Constructive Approximation , 1993, Grundlehren der mathematischen Wissenschaften.

[21]  Pencho Petrushev,et al.  Irregular sampling of band-limited functions on the sphere , 2014 .

[22]  D. S. Seljebotn.  WAVEMOTH–FAST SPHERICAL HARMONIC TRANSFORMS BY BUTTERFLY MATRIX COMPRESSION , 2011, 1110.4874.

[23]  Leopold Fejér,et al.  On the infinite sequences arising in the theories of harmonic analysis, of interpolation, and of mechanical quadratures , 1933 .

[24]  Yuan Xu,et al.  Decomposition of spaces of distributions induced by tensor product bases , 2009, 0902.2601.

[25]  Mark Tygert,et al.  Fast algorithms for spherical harmonic expansions, III , 2009, J. Comput. Phys..

[26]  C. W. Clenshaw,et al.  A method for numerical integration on an automatic computer , 1960 .

[27]  Pencho Petrushev,et al.  Decomposition of Besov and Triebel–Lizorkin spaces on the sphere , 2006 .

[28]  Pencho Petrushev,et al.  Localized Tight Frames on Spheres , 2006, SIAM J. Math. Anal..

[29]  L. Fejér,et al.  Mechanische Quadraturen mit positiven Cotesschen Zahlen , 1933 .

[30]  Paul Nevai,et al.  Distribution of zeros of orthogonal polynomials , 1979 .