Strictly positive definiteness of Hermite interpolation on spheres
Let \(S^{m-1}\) be the unit sphere in the Euclidean space and \(h(x,y)\) be a spherical kernel, \(h(x,y)=\sum^\infty_{k=0} \widehat{h}_kP^\lambda_k(xy)\), where \(\lambda=(m-2)/2\), \(P^\lambda_k\) is the Gegenbauer polynomial. As an extension of the paper by \textit{Y. Xu} and \textit{W. Cheney} [Proc. Am. Math. Soc. 116, No. 4, 977-981 (1997; Zbl 0787.43005)] sufficient conditions were found for positive definite kernels \(h\) to be strictly positive definite and hence well-posed for Hermite scattered data interpolation on Euclidean unit spheres. These sufficient conditions involve the value of \(\widehat{h}_k\).
scientific article; zbMATH DE number 1309404