Ideal Interpolation: Translations to and from Algebraic Geometry

The discretization for bivariate ideal interpolation

“…Suppose I n is the ideal determined by δ (0,0) • L n (D), M x and M y are the corresponding multiplication matrices. The following result can be obtained by [3,25], we give its proof here for completeness.…”

“…In [16], de Boor and Shekhtman gave a linear algebra proof for the bivariate case. Afterwards, Shekhtman summarized this proof in his survey [3]. Note that Shekhtman discusses the discretization problem from the aspect of interpolation space while we start from the space of interpolation condition functionals.…”

“…(6) andã i (t 0 ). The following table records the results: It is easy to see thatã 0 (t) = 0, and we can computẽ a 1,2 (t) := RationalInterpolation([ [1,1], [2,4]], t) = t 2 , a 1,3 (t) := RationalInterpolation([ [1,1], [2,4], [3,9]], t) = t 2 using Maple, thusã 1 (t) = t 2 .ã 2 (t) = (1 − t)/t can be obtained in the same way. Forã 3 (t), we need to computẽ…”

“…Note that such an A(t) is a nonderogatory matrix (see [3,Theorem 6.2.13]) and A(t) → M y as t tends to zero.…”

“…de Boor and Shekhtman have made significant progress in multivariate polynomial interpolation following this concept [2,3]. A polynomial subspace is said to be D-invariant if it is closed under differentiation.…”