Polynomial Bases on the Sphere

Polynomial interpolation on the unit sphere II

“…In this sense, the present paper closes the gap left in [13]. In comparison to [6], the factorization method allows to obtain more sets of points that solve problem 1.…”

“…For the simplest case n = 2m, the set of (2m + 1) 2 points lie on 2m + 1 latitudes, each of them containing 2m + 1 equally spaced nodes. In [6], another family of points that solves problem 1 was found, for which n = 2m − 1. There the points lie on 2m latitudes and each latitude has an even number of 2m equally spaced points.…”

“…This problem has been studied recently by several authors. In the context of the present paper, the following references might be of interest: [4,6,7,10,13,14].…”

“…In this case, the 2m latitudes are divided into two groups; the equidistant points on one half of the latitudes need to differ by a rotation from the points on the other half of the latitudes. While the proof in [6] is based on the analysis of the determinants of the interpolation matrix, the proof in [13] uses a factorization method which avoids the determinants. Furthermore, the factorization method provides many more sets of points leading to poised problems.…”

Rotationally invariant quadratures for the sphere

“…In the resulting representation, the coefficients, up to a factor, are the values on the grid of the function being represented. We may interpret this construction as an analogue of Lagrange interpolation on the sphere (see also Sloan 1995;Sloan & Womersley 2000) and note that it allows us to develop well-conditioned linear systems for interpolation in contrast to some earlier constructions (Fernandez 2005;Fernandez & Prestin 2005). …”