PhD Thesis

My PhD Thesis concerns certain topics in polynomial interpolation theory and was carried out under the supervision of Ragni Piene and Ragnar Winther. The defense was on the 29th of March 2011. Below you can find PDF files of the trial lecture, the defense talk, and my thesis.

  • PhD Thesis: Topics in Polynomial Interpolation Theory (PDF)

      • Summary: This thesis was carried out at the Centre of Mathematics for Applications at the University of Oslo. Two topics were studied with the goal to improve methods for finding smooth curves and surfaces passing through -- or interpolate -- certain known points. Such methods are used in a variety of contexts where shapes are constructed from discrete data, from typography to the computer-aided geometric design of cars, ships, and airplanes. The special case where the curves and surfaces are represented locally by polynomials is particularly popular in industrial applications. The numerical properties of such interpolants depend very much on how the data points are situated in space. The simplest configurations of data points are the rectangular grids, which are the points of intersection of some collection of horizontal and vertical lines. The triangular grids form another classical type of configurations; they are the points in the plane with nonnegative integral coordinates (i, j) with i+j bounded from above by some fixed integer. Because of their convenient properties, both configurations have been studied extensively in the literature. The first part of the thesis sets forth explicit formulas for the coefficients of polynomial interpolants to implicit functions on rectangular grids. A closed formula for the higher-order derivatives of implicit functions appears as a limiting case of these formulas. The second part delves into certain generalizations of triangular grids -- generalized principal lattices -- that are well suited for interpolation. Applying the theory of real algebraic curves then allows the construction of many new examples of such configurations.

  • Trial Lecture: Kergin Interpolation, Divided Differences, and Simplex Splines (PDF)

  • Defense Talk: Topics in Polynomial Interpolation Theory (PDF, Sage Worksheet)

My brother-in-law, Gaute Thøgersen, made this video.