Constructive approximations of spherical functions

Approximation of functions on the sphere arises in almost all applications modeling data collected on the surface of the earth and for reconstruction of various processes in spherical coordinates. Constructive approximation of high dimensional spherical functions are useful for approximation of processes of several variables on a compact subset of a Euclidean space by mapping the data onto the unit sphere of a space having one higher dimension, avoiding the boundary effect of the set. Interpolation operators on the circle and periodic domains (based on a class of basis functions and data values) are essential for many high performance simulations. These operators are represented by analytical summation formulas that can be computed very efficiently using the fast Fourier transform. This work is concerned with construction of a similar class of interpolation and quasi-interpolation operators on the unit sphere in ℝ^q , for q = 3, 4, 5, · · ·, using a new class of basis functions. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)