Zeros of orthogonal polynomials

Let $$P$$
 be a set of $$n$$
 points in the plane, not all on a line We show that if $$n$$
 is large then there are at least $$n/2$$
 ordinary lines, that is to say lines passing through exactly two points of $$P$$
  This confirms, for large $$n$$
 , a conjecture of Dirac and Motzkin In fact we describe the exact extremisers for this problem, as well as all sets having fewer than $$n-C$$
 ordinary lines for some absolute constant $$C$$
  We also solve, for large $$n$$
 , the “orchard-planting problem”, which asks for the maximum number of lines through exactly 3 points of $$P$$
  Underlying these results is a structure theorem which states that if $$P$$
 has at most $$Kn$$
 ordinary lines then all but O(K) points of $$P$$
 lie on a cubic curve, if $$n$$
 is sufficiently large depending on $$K$$
 

/pdf/on-sets-defining-few-ordinary-lines-3yaoygcdj2.pdf

On Sets Defining Few Ordinary Lines

GC-sets are subsets T of $\mathbb{R}^d$ of the appropriate cardinality $\dim\Pi_n$ for which, for each τ ∈ T, there are n hyperplanes whose union contains all of T except for τ, thus making interpolation to arbitrary data on T by polynomials of degree ≤ n uniquely possible. The existing bivariate theory of such sets is extended to the general multivariate case and the concept of a maximal hyperplane for T is highlighted, in hopes of getting more insight into existing conjectures for the bivariate case.

/pdf/multivariate-polynomial-interpolation-conjectures-concerning-58gut8u9lc.pdf

Multivariate polynomial interpolation: conjectures concerning GC-sets

Principal lattices are classical simplicial configurations of nodes suitable for multivariate polynomial interpolation in n dimensions. A principal lattice can be described as the set of intersection points of n + 1 pencils of parallel hyperplanes. Using a projective point of view, Lee and Phillips extended this situation to n + 1 linear pencils of hyperplanes. In two recent papers, two of us have introduced generalized principal lattices in the plane using cubic pencils. In this paper we analyze the problem in n dimensions, considering polynomial, exponential and trigonometric pencils, which can be combined in different ways to obtain generalized principal lattices.We also consider the case of coincident pencils. An error formula for generalized principal lattices is discussed.

Interpolation lattices in several variables

Under general conditions, the equation g(x, y) = 0 implicitly defines y locally as a function of x. In this article, we express divided differences of y in terms of bivariate divided differences of g, generalizing a recent result on divided differences of inverse functions.

https://www.duo.uio.no/bitstream/handle/10852/10725/dravhandling-muntingh.pdf?sequence=1

Topics in Polynomial Interpolation Theory

Geometric characterization and generalized principal lattices

Given a cubic pencil, an addition of lines can be defined in order to construct generalized principal lattices. In this paper we show the converse: the lines defining a generalized principal lattice belong to the same cubic pencil, which is unique for degrees ≥ 4.

Generalized principal lattices and cubic pencils

The Lagrange interpolation problem on spaces of symmetric bivariate polynomials is considered to reduce the interpolation problem to problems of approximately half dimension. The Berzolari-Radon construction is adapted to these kinds of problems by considering nodes placed on symmetric lines or symmetric pairs of lines. A Newton formula for the symmetric interpolant using the Berzolari-Radon construction is proposed.

Interpolation with symmetric polynomials

Bivariate polynomials defined on the unit disk are used to reconstruct a wavefront from a data sample. We analyze the interpolation problem arising in critical sampling, that is, using a minimal sample. The interpolant is expressed as a linear combination of Zernike polynomials, whose coefficients represent relevant optical features of the wavefront. We study the propagation of errors of the polynomial values and their coefficients, obtaining bounds for the Lebesgue constants and condition numbers. A node distribution leading to low Lebesgue constants and condition numbers for degrees up to 20 is proposed.

Carmen Godés

Papers

Geometric characterization and generalized principal lattices

Generalized principal lattices and cubic pencils

Interpolation with symmetric polynomials

Interpolation on the disk

Configurations of nodes with defects greater than three