On the Lebesgue constant of the Morrow-Patterson points

TL;DR

This work establishes that the Lebesgue constant for interpolation at the Morrow-Patterson points $MP_n$ on the square grows quadratically with the degree, $\Lambda_n^{MP}=\mathcal{O}(n^2)$, improving previous bounds and corroborating prior numerical evidence. It presents three independent constructions of $MP_n$ (Lissajous-curve, zeros of bivariate orthogonal polynomials, and interlacing rectangular grids), derives an exact cubature formula for the weight $\sqrt{1-x^2}\sqrt{1-y^2}$ with weights $\omega_{m,k}=C_n(1-x_m^2)(1-y_k^2)$, and formulates the interpolation operator in terms of an orthonormal basis built from Chebyshev polynomials of the second kind. The paper also introduces optimal admissible meshes based on MP$_{\nu}$ and related meshes, providing sharp Lebesgue-function bounds and a framework to transfer node-set quality to stability via discrete/integral norm comparisons. Numerical experiments with the Matlab implementation $\texttt{Leb\_MP.m}$ confirm the theoretical $\mathcal{O}(n^2)$ growth and illustrate near-optimal behavior relative to the quadratic benchmarks, while leaving open precise edge/corner extremal behavior and certain symmetry questions for future work.

Abstract

The study of interpolation nodes and their associated Lebesgue constants are central to numerical analysis, impacting the stability and accuracy of polynomial approximations. In this paper, we will explore the Morrow-Patterson points, a set of interpolation nodes introduced to construct cubature formulas of a minimum number of points in the square for a fixed degree $n$. We prove that their Lebesgue constant growth is ${\cal O}(n^2)$ as was conjectured based on numerical evidence about twenty years ago in the paper by Caliari, M., De Marchi, S., Vianello, M., {\it Bivariate polynomial interpolation on the square at new nodal sets}, Appl. Math. Comput. 165(2) (2005), 261--274.

Figures (5)

• Figure 1: Morrow-Patterson and Padua points for $n=6$ and the corresponding Lissajous curves
• Figure 2: The MP points as two interlacing rectangular grids with $n=2$: $K_1$ is the black rectangle, $K_2$ is the blue rectangle
• Figure 3: The numerically evaluated Lebesgue constant of interpolation at the Morrow-Patterson points (log scale) compared with $(0.7n+1)^2$ and $(0.75n+1)^2$.
• Figure 4: Top: the Lebesgue function for $n=30$. Bottom: the behavior of the Lebesgue constant and the bound $(n+1)(n+2)/2$.
• Figure 5: Extended Morrow-Patterson points. Top: the Lebesgue function for $n=30$. Bottom: the behavior of the Lebesgue constant compared with the bound $(n+1)(n+2)/2$.

Theorems & Definitions (24)

• Remark 1
• Theorem 1
• Proposition 1
• Proposition 2
• proof
• Theorem 2
• proof
• Remark 2
• Lemma 1
• proof