On the approximation properties of fast Leja points

TL;DR

The paper addresses whether fast Leja points on an interval yield good univariate polynomial interpolation. It introduces Property ($\star$), an asymptotic distribution condition for interpolation arrays, and proves that if ($\star$) holds for the fast Leja set on $I=[0,1]$, then $\lim_{n\to\infty} |VDM(a_1,\dots,a_n)|^{2/[n(n+1)]}=d(I)$, establishing the array as good for interpolation via the BBCL framework. It also provides general sufficient conditions ensuring ($\star$), connects the construction to pseudo-Leja and $\tau$-Leja concepts, and offers conjectures and numerical evidence supporting fast Leja points as $\tau$-Leja with $\tau\in(\tfrac{1}{2},1)$. The results pave a rigorous path to validating fast Leja points for high-quality Lagrange interpolation and hint at extensions to unions of intervals or curves.

Abstract

Fast Leja points on an interval are points constructed using a discrete modification of the algorithm for constructing Leja points. Not much about fast Leja points has been proven theoretically. We present an asymptotic property of a triangular interpolation array, and under the assumption that fast Leja points satisfy this property, we prove that they are good for Lagrange interpolation.

Figures (3)

• Figure 1: Illustration of the bounds (valid for all $k$) $B_1|(j-k)/n|^{\alpha_1} \leq |b_{nj}-b_{nk}| \leq B_2|(j-k)/n|^{\alpha_2}$ when $j=58$, for a randomly generated distribution of 100 points in the interval $[0,1]$. Here $0.5<\alpha_2<1<\alpha_1<1.5$.
• Figure 2: Illustration of the estimates in Theorem \ref{['thm:generalthm']} for a 50-point approximation to the equilibrium distribution for $[0,1]$ with $j=5$, and $0.5<\alpha_2<1<\alpha_1<1.5$. The lower estimate only holds outside a (small) neighborhood of the end points.
• Figure 3: A plot of the graph $(x,p_{13}(x))$, where the zeros of $p_{13}$ are the first 13 fast Leja points on $[0,1]$. The points $(m_j,2p_{13}(m_j))$ are indicated by circles, where $m_j$ is the midpoint of the $j$-th interval between adjacent fast Leja points. These midpoints are the candidates for the next fast Leja point.

Theorems & Definitions (17)

• Theorem : cf. Theorem \ref{['thm:d']}
• Lemma 2.1
• proof
• Proposition 2.2
• proof
• Example 3.1
• Theorem 3.2
• proof
• Theorem 3.3
• proof