Multivariate Hermite interpolation by algebraic polynomials: A survey

Recent developments and open problems in linear series

“…, ∂ α N x ∂ β N y } which (interpreting (α i , β i ) as points in N 2 ) is closed under downward and leftward moves; in the literature such sets E are known as staircases (see [15]) and they are used to define monomial ideals I(E) = (x α y β ) (α,β) ∈E . In the method above one then has to use a modified jet sheaf where m Such a generalization allows to deal with the Hermite interpolation schemes of tree type considered by Lorentz, (see [37], Section 3), which is actually a specialized case of the general monomial interpolation problem suggested as Problem 2.4.2. Note that in the original problem the coordinates in which the scheme is monomial can be "deformed" by the immersion of the scheme in the surface, whereas here they are torically fixed.…”

A chronology of interpolation: from ancient astronomy to modern signal and image processing

“…23 Hermite and Birkhoff type of interpolation problems-and their multivariate versions, not necessarily on Cartesian grids-have received much attention in the past decades. A more detailed treatment is outside the scope of this paper, however, and the reader is referred to relevant books and reviews [70], [79]- [83]. 21 Many of them introduced their own system of notation and terminology, leading to confusion and researchers reformulating existing results.…”

A case of multivariate Birkhoff interpolation using high order derivatives

“…For a prescribed type of Hermite or Birkhoff interpolation problem, find the conditions by which there exists a unique solution. This problem is central in Approximation Theory (see [18,22,4] and references therein). It was traditionally considered, from a somewhat different point of view, also in Algebraic geometry (see [20,1,8,9,13] and references therein).…”