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

Abstract

Let $Γ$ be an $L$-shape arc consisting of 2 line segments that meet at an angle different from $π$ in the complex $z$-plane $\CC$. This paper is to investigate the behavior of the polynomial interpolants at the Fejér points, defined by $\{z_{n,k} = ψ(e^{i(2kπ+ θ)/(n+1)})\}$ for any choice of $θ$. In this regard, we recall that for the interval [-1, 1], the Fejér points $\{z_{n,k} = ψ^*(e^{i(2k+1)π/(n+1)})\}$ 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 $Γ$, 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 $n^β$, for some $β>0$ that depends on $p >1$.