Polynomial Interpolation to Data on Flats in Rd Original Research Article

Stimulated by recent work of Hakopian and Sahakian, polynomial interpolation to data at all the s-dimensional intersections of an arbitrary sequence of hyperplanes in Rd is considered, and reduced, by the adjunction of an additional s hyperplanes in general position with respect to the given sequence, to the case s=0 solved much earlier by two of the present authors. In particular, interpolation is from the very same polynomial spaces already used earlier. The difficult question of multiplicity and corresponding matching of derivative information is completely solved, with the number of independent derivative conditions at an intersection exactly equal to that intersectionʹs multiplicity. Also, the consistency requirements placed on the data are minimal in the sense that they need to be checked only at the finitely many 0-dimensional intersections of the hyperplanes involved. The arguments used provide, incidentally, further insights into the two polynomial spaces, P(Ξ) and D(Ξ), of basic interest in box spline theory.