Norm estimates of interpolation matrices and their inverses associated with strictly positive definite functions

Abstract

Abstract. In this paper, we estimate the norms of the interpolation matrices and their inverses that arise from scattered data interpolation on spheres with strictly positive definite functions. §1. Introductionwhere xy denotes the usual inner product ofwhere H n is the space of spherical harmonics of degree n. Orthogonality here is with respect to the inner productwhere ω m−1 is the rotation invariant measure on the sphere, normalized so that. . , Y n,An } be an orthonormal basis for H n , where A n = dim H n .A continuous function g : [0, π] → R is said to be positive definite on S m−1 if for any N points x 1 , x 2 , . . . , x N ∈ S m−1 , the matrix A having elementsis non-negative definite. If for any N distinct points x 1 , x 2 , . . . , x N ∈ S m−1 , the matrix A is positive definite, then the function g is said to be strictly positive definite on S m−1 .Schoenberg [S] characterized the class of all positive definite functions on S m−1 as those that have the formwhere a n ≥ 0 and ∞ n=0 a n < ∞, and P (λ) n denotes the Gegenbauer (ultraspherical) polynomials normalized so that P (λ) n (1) = 1.

Jackson kernels: a tool for analysing the decay of eigenvalue sequences of integral operators on the sphere

“…defines families {Z m n,t } of locally supported kernels on S m . A construction of such kernels by iteration with spherical convolutions can be found in [11,19]. Since…”

Interpolation on the sphere: a fast solution technique

“…Error analysis for this problem was carried out by Narcowich et al [9,10]. Levesley et al [7] to show that the matrix A is ill-conditioned. In the following section, we design a preconditioner for this system, using multiplicative Schwarz methods.…”

“…Thanks to (7) we view the interpolation problem (1) as a variational problem, with a bilinear form defined on H τ (S) by a(v, w) := v, w φ for all v, w ∈ H τ (S) .…”

“…Both theory [7] and numerical experiments (Section 4) show that the matrices arising from this interpolation problem are very ill-conditioned. We present an algorithm which uses spherical radial basis functions to construct the interpolant and a multiplicative Schwarz preconditioner to accelerate the solution process.…”