Polynomial interpolation on the unit sphere II

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 proved to have a unique solution for several sets of points. The points are located on a number of circles on the sphere with even number of points on each circle. The proof is based on a method of factorization of polynomials.

Polynomial interpolation on the unit sphere and some properties of its integral means

“…In particular, Corollary 3.2 in [19] points out that, when n = 2m, the set A consisting of (2m + 1) 2 points lying on 2m + 1 latitudes, each of them contains 2m + 1 equidistant points, solves the problem. In [10], the authors gave an analogous result, but each parallel circle contains an even number of points. To investigate the problem, the authors use the spherical coordinates…”

“…The aim of this paper is to investigate Hermite interpolation on the unit sphere. Our study is inspired from [10,19] in which the authors studied the following problem of Lagrange interpolation on S: Problem. Let A = {a i : 1 ≤ i ≤ (n + 1) 2 } be a set of distinct points on S. Find conditions on A such that there is a unique polynomial p ∈ P n (S) satisfying…”

“…Then they proved factorization theorems and used them to solve the problem. In this paper, we generalize [19,Corollary 3.2] and a special case of [10,Theorem 3.1] in which all latitudes contain the same number of points. Here, Lagrange interpolation is extended to Hermite interpolation that can be viewed as a result of coalescing latitudes with respect to the variable θ.…”

“…The first main results are Theorem 3.2 (with odd number of points on each circle) and Theorem 3.5 (with even number of points on each circle). To prove the theorems, we modify the methods given in [10,19]. Some arguments in [10,19] are repeated suitably in this paper.…”

“…To prove the theorems, we modify the methods given in [10,19]. Some arguments in [10,19] are repeated suitably in this paper. Next we wish to study the continuity and convergence properties of the Hermite interpolation polynomials.…”