On Marcinkiewicz-Zygmund inequalities and $A_p$-weights for $L$-shape arcs

TL;DR

This paper investigates polynomial interpolation on L-shaped arcs, revealing that Lebesgue constants grow logarithmically and Marcinkiewicz-Zygmund inequalities have polynomial growth, highlighting differences from classical interval cases.

Contribution

It extends Marcinkiewicz-Zygmund inequalities and $A_p$-weight analysis to L-shaped arcs, showing their distinct behavior compared to interval cases.

Findings

Lebesgue constants grow as O((log n)^2) for L-shaped arcs.

$A_p$-weight conditions do not extend from [-1,1] to L-shaped arcs.

Marcinkiewicz-Zygmund inequalities grow at rate n^β, with β > 0.

Abstract

Let $\Gamma$ be an $L$-shape arc consisting of 2 line segments that meet at an angle different from $\pi$ in the complex $z$-plane $\CC$. This paper is to investigate the behavior of the polynomial interpolants at the Fej\'er points, defined by $\{z_{n,k} = \psi(e^{i(2k\pi + \theta)/(n+1)})\}$ for any choice of $\theta$. In this regard, we recall that for the interval [-1, 1], the Fej\'er points $\{z_{n,k} = \psi^*(e^{i(2k+1)\pi/(n+1)})\}$ agree with the Chebyshev points and that the Chebyshev points are most commonly used as nodes for Lagrange polynomial interpolation. On the other hand, numerical experimentation demonstrates that for a typical open $L$-shape arc $\Gamma$, the Lebesgue constants tend to $\infty$ at the rate of $O((log(n))^2)$, as the polynomial degree $n$ increases, while the $A_{p}$-weight conditions for the Fej\'er points $\{z_{n,k}\}$ do not carry over from [-1, 1] to a truly $L$-shape arc. Further numerical experiments also demonstrate that the least upper bounds of the Marcinkiewicz-Zygmund inequalities for the canonical Lagrange interpolation polynomials at $\{z_{n,k}\}$ seem to grow at the rate of $n^{\beta}$, for some $\beta >0$ that depends on $p >1$.