Polynomial Interpolation on the Unit Sphere

Abstract

The problem of interpolation at (n + 1) 2 points on the unit sphere S 2 by spherical polynomials of degree at most n is studied. Many sets of points that admit unique interpolation are given explicitly. The proof is based on a method of factorization of polynomials. A related problem of interpolation by trigonometric polynomials is also solved.

Polynomial interpolation on the unit sphere II

“…However, the new interpolation problem is different from the one with an odd number of points on each latitude and has to be solved using a completely different method. 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.…”

“…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 [13] a large family of sets of interpolation points is given explicitly, each set solving problem 1. Let us briefly describe this construction.…”

“…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.…”

“…Furthermore, the factorization method provides many more sets of points leading to poised problems. A key observation in [13] is that the use of equidistant points allows us to reduce the problem on the sphere to a special trigonometric interpolation problem.…”