On the poisedness of Bojanov–Xu interpolation

Abstract

In this paper, we consider the bivariate Hermite interpolation introduced by Bojanov and Xu [SIAM J. Numer. Anal. 39(5) (2002) 1780-1793. The nodes of the interpolation with 2k− , where = 0 or 1, are the intersection points of 2k+1 distinct rays from the origin with a multiset of k+1− concentric circles. Parameters are the values and successive radial derivatives, whenever the corresponding circle is multiple. The poisedness of this interpolation was proved only for the set of equidistant rays [Bojanov and Xu, 2002] and its counterparts with other conic sections [Hakopian and Ismail, East J. Approx. 9 (2003) 251-267]. We show that the poisedness of this (k + 1 − )(2k + 1) dimensional Hermite interpolation problem is equivalent to the poisedness of certain 2k + 1 dimensional Lagrange interpolation problems. Then the poisedness of Bojanov-Xu interpolation for a wide family of sets of rays satisfying some simple conditions is established. Our results hold also with above circles replaced by ellipses, hyperbolas, and pairs of parallel lines.Next a conjecture [Hakopian and Ismail, J. Approx. Theory 116 (2002) 76-99] concerning a poisedness relation between the Bojanov-Xu interpolation, with set of rays symmetric about x-axis, and certain univariate lacunary interpolations is established. At the end the poisedness for a wide class of lacunary interpolations is obtained.

Taylorian points of an algebraic curve and bivariate Hermite interpolation

“…The works of Lorenz [14,15] is more particularly dedicated to Hermite interpolation. Recent interesting results around Bojanov-Xu schemes include [1,11,12]. The remarkable work of de Boor and Ron [7,8] deserves particular attention.…”