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

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 Fejér 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 Fejér points $\{z_{n,k}={\psi }^{\ast }(e^{i(2k+1)\pi /(n+1)})\}$ 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 Fejér 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 $p>1$.

16 pages. 7 figures and 3 tables. Keywords:Marcinkiewicz-Zygmund inequalities; $A_p$ weight; Lebesgue constants. J. Geom. Anal. (2021)

References in corpus (1)

• Marcinkiewicz-Zygmund inequalities