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The Fekete problem in segmental polynomial interpolation

Ludovico Bruni Bruno, Wolfgang Erb

Abstract

In this article, we study the Fekete problem in segmental and combined nodal-segmental univariate polynomial interpolation by investigating sets of segments, or segments combined with nodes, such that the Vandermonde determinant for the respective polynomial interpolation problem is maximized. For particular families of segments, we will be able to find explicit solutions of the corresponding maximization problem. The quality of the Fekete segments depends hereby strongly on the utilized normalization of the segmental information in the Vandermonde matrix. To measure the quality of the Fekete segments in interpolation, we analyse the asymptotic behaviour of the generalized Lebesgue constant linked to the interpolation problem. For particular sets of Fekete segments we will get, similar to the nodal case, a favourable logarithmic growth of this constant.

The Fekete problem in segmental polynomial interpolation

Abstract

In this article, we study the Fekete problem in segmental and combined nodal-segmental univariate polynomial interpolation by investigating sets of segments, or segments combined with nodes, such that the Vandermonde determinant for the respective polynomial interpolation problem is maximized. For particular families of segments, we will be able to find explicit solutions of the corresponding maximization problem. The quality of the Fekete segments depends hereby strongly on the utilized normalization of the segmental information in the Vandermonde matrix. To measure the quality of the Fekete segments in interpolation, we analyse the asymptotic behaviour of the generalized Lebesgue constant linked to the interpolation problem. For particular sets of Fekete segments we will get, similar to the nodal case, a favourable logarithmic growth of this constant.
Paper Structure (18 sections, 14 theorems, 84 equations, 4 figures, 1 table)

This paper contains 18 sections, 14 theorems, 84 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Interpolation operators
  3. The nodal interpolator
  4. The segmental interpolator
  5. A combined nodal-segmental interpolator based on function averages
  6. Lebesgue constants
  7. The nodal Lebesgue constant
  8. The segmental Lebesgue constant
  9. The Lebesgue constant for mixed nodal-segmental interpolation
  10. Fekete problems
  11. The Lagrange basis functions for Fekete nodes and segments
  12. The nodal Fekete problem
  13. The non-normalized Fekete problem for concatenated segments
  14. The normalized Fekete problem for nodal-segmental interpolation
  15. The case (C1): concatenated segments
  16. ...and 3 more sections

Key Result

Proposition 1

Let $\{ s_i \}_{i=1}^{r}$ be a unisolvent collection of disjoint segments and let $\{ \widehat{\ell}_{s_i} \}_{i=1}^{r}$ be the corresponding Lagrange basis polynomials. Then As a consequence,

Figures (4)

  • Figure 1: Lebesgue constants for concatenated (non-normalized) Fekete segments with degrees up to $r = 150$. The figures indicate a sublinear growth of the Lebesgue constant proportional to $\sqrt{r}$.
  • Figure 2: Segmental interpolation for Fekete segments in the class (C1) for the non-normalized Fekete (left) and the normalized Fekete problem (right). In the normalized case, the first and the last segment degenerate to single nodes.
  • Figure 3: Comparison of the Lebesgue constants for Fekete segments in the classes (C1) and (C2). Left: comparison between normalized and non-normalized (NN) Fekete segments in the class (C1). Right: comparison between the Fekete segments in the class (C1) and the class (C2) for the values $\lambda = 0.5$ and $\lambda = 0$ (the nodal case).
  • Figure 4: Interpolation for Fekete segments in the class (C2) with $\lambda = 0.5$ (left) and $\lambda = 0.2$ (right).

Theorems & Definitions (18)

  • Proposition 1
  • Remark 2
  • Proposition 3
  • Definition 4
  • Theorem 5
  • Remark 6
  • Lemma 7
  • Lemma 8
  • Lemma 9
  • Lemma 10
  • ...and 8 more