Transfinite mean value interpolation in 3D

s for MAIA 2007 The limits of bivariate Lagrange projectors Carl de Boor∗ and Boris Shekhtman Eastsound, Washington In a talk in Norway in 2003, the first author conjectured that any finite-rank ideal projector (i.e., finite-rank linear projector on the space of polynomials in d variables whose kernel is a polynomial ideal, i.e., a linear space also closed under pointwise multiplication by polynomials) is the (pointwise) limit of Lagrange projectors (i.e., projectors whose kernel consists of all polynomials that vanish on a certain finite set). The chair of that talk’s session, Geir Ellingsrud, pointed out to that first author (afterwards :-) that this conjecture must be wrong for d > 2 but may be correct for d = 2, and both these statements were subsequently proved by the second author, using tools from algebraic geometry. The two authors are therefore pleased to present here a proof of the conjecture for d = 2 that uses nothing more than linear algebra. Transfinite mean value interpolation in 3D Solveig Bruvoll∗ and Michael Floater Oslo, Norway In this talk we study mean value interpolation over volumetric domains of arbitrary topology. We derive conditions on the boundary of the domain to guarantee interpolation when the data is continuous. By deriving the normal derivative of the interpolant and of a mean value weight function we construct a transfinite Hermite interpolant.