This paper considers the variational problem of Hermite interpolation and its error bounds. The optimal Hermite interpolant, which minimises the semi-norm of the reproducing kernel Hilbert space C h determined by given r-CPD m function h, is just the h-spline Hermite interpolant. The results on error estimation and convergence rate of the h-spline interpolant generalise those of W.

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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.

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