This is a survey of the main results on multivariate polynomial interpolation in the last twenty five years, a period of time when the subject experienced its most rapid development. The problem is considered from two different points of view: the construction of data points which allow unique interpolation for given interpolation spaces as well as the… (More)
Principal lattices in the plane are distributions of points particularly simple to use Lagrange, Newton or Aitken-Neville interpolation formulas. Principal lattices were generalized by Lee and Phillips, introducing three-pencil lattices, that is, points which are the intersection of three lines, each one belonging to a different pencil. In this… (More)
Principal lattices are distributions of points in the plane obtained from a triangle by drawing equidistant parallel lines to the sides and taking the intersection points as nodes. Interpolation on principal lattices leads to particularly simple formulae. These sets were generalized by Lee and Phillips considering three-pencil lattices, generated by three… (More)
In this paper we study multivariate polynomial interpolation on Aitken–Neville sets by relating them to generalized principal lattices. We express their associated divided differences in terms of spline integrals. Dedicated to Günter Mühlbach on occasion of his 65th birthday.
A nonsingular matrix is called almost strictly totally positive when all its minors are nonnegative, and furthermore these minors are positive if and only if their diagonal entries are positive. In this paper we give a characterization of these matrices in terms of the positivity of a very reduced number of their minors (which are called boundary minors),… (More)
In 1977 Chung and Yao introduced a geometric characterization in multivariate interpolation in order to identify distributions of points such that the Lagrange functions are products of real polynomials of first degree. We discuss and describe completely all these configurations up to degree 4 in the bivariate case. The number of lines containing more nodes… (More)
One of the problems in bivariate polynomial interpolation is the choice of a space of polynomials suitable for interpolating a given set of data. Depending on the number of data, a usual space is that of polynomials in 2 variables of total degree not greater than k. However, these spaces are not enough to cover many interpolation problems. Here, we are… (More)