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 $p >1$.
Categories: math.CA