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 \({\mathbb C}\). Application of the exterior conformal map \(\psi \) from \(|w| > 1\) onto \({\mathbb C}\backslash \Gamma \), with \(\psi (\infty )= \infty \), introduces the level curves \(\Gamma _n=\{z= \psi (w):|w|=1+{1\over {n+1}}\}\). Let \(\psi ^*\) denote the continuous extension of \(\psi \) from \(|w|> 1\) to \(|w|\ge 1\), so that any family \(\{z_{n,k}: k = 0, 1, \dots , n\}\) of points on \(\Gamma \) can be written as \(\{z_{n,k} = \psi ^*(w_{n,k})\}\), where \(|w_{n,k}|= 1\). Let \(\omega _n (z)= \Pi ^n_{k=0} (z-z_{n,k})\). The main objective of this paper is to show that for \(L\)-shape arcs, validation of the Marcinkiewicz–Zygmund inequalities is equivalent to that of the totality of the \(A_p\)-weight conditions of \(|\omega _n (z) |\) on \(\Gamma _n\) and a mild separation condition of \(\{z_{n,k}\}\). Since the Marcinkiewicz–Zygmund inequalities are essential to the study of Lagrange polynomial interpolation of continuous functions at the nodes \(\{z_{n,k}\}\), another objective of this paper is to investigate the behavior of the polynomial interpolants at the Fejér points, defined by \(\{z_{n,k} = \psi ^*(\mathrm{{e}}^{i(2k\pi + \theta )/(n+1)})\}\) for any choice of \(\theta \). In this regard, we recall that for the interval [\(-1\), 1], the Fejér points \(\{z_{n,k} = \psi ^*(\mathrm{{e}}^{i(2k+1)\pi /(n+1)})\}\) 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 \(p >1\).

让\（\伽玛\）是\（L \） -形弧由2个线段的，在从一个角度不同满足\（\ PI \）在复\（Z \） -平面\（{\ mathbb C} \）。外部保形映射的应用\（\ PSI \）从（| W |> 1 \）\到\（{\ mathbbÇ} \反斜杠\伽玛\） ，用\（\ PSI（\ infty）= \ infty \ ），介绍电平曲线\（\ Gamma _n = \ {z = \ psi（w）：| w | = 1 + {1 \ {n + 1}} \} \）。令\（\ psi ^ * \）表示\（\ psi \）从\（| w |> 1 \）到\（| w | \ ge 1 \）的连续扩展，因此\（\ Gamma \）上点的任何族\（\ {z_ {n，k}：k = 0，1，\ dots，n \} \）都可以写成\（\ {z_ {n ，k} = \ psi ^ *（w_ {n，k}）\} \），其中\（| w_ {n，k} | = 1 \）。令\（\ omega _n（z）= \ Pi ^ n_ {k = 0}（z-z_ {n，k}）\）。本文的主要目标是表明，\（L \） -形弧，Marcinkiewicz型-上Zygmund不等式验证是等同于的总体的\（A_p \）的-weight条件\（| \欧米加_n（z）| \）放在\（\ Gamma _n \）上，且条件为\（\ {z_ {n，k} \} \）\。由于Marcinkiewicz-Zygmund不等式对于研究节点\（\ {z_ {n，k} \} \）上连续函数的Lagrange多项式插值至关重要，因此，本文的另一个目标是研究多项式插值的行为在Fejér点处，由\（\ {z_ {n，k} = \ psi ^ *（\ mathrm {{e}} ^ {i（2k \ pi + \ theta）/（n + 1）}）定义\ } \）用于\（\ theta \）的任何选择。在这方面，我们记得对于间隔[ \（-1 \），1]，Fejér点\（\ {z_ {n，k} = \ psi ^ *（\ mathrm {{e}} ^ {i （2k + 1）\ pi /（n + 1）}）\} \）同意Chebyshev点，并且Chebyshev点最常用作Lagrange多项式插值的节点。在另一方面，数值实验证明，对于一个典型的开放\（L \） -形弧\（\伽玛\） ，勒贝格常数趋于\（\ infty \）在速率\（O（（\日志（n））^ 2）\），随着多项式次数\（n \）的增加，而Fejér点的\（A_ {p} \）-权重条件\（\ {z_ {n，k} \} \）不要从[ \（-1 \），1]延续到真正的\（L \）形弧。进一步的数值实验还表明，在\（\ {z_ {n，k} \} \）处的规范Lagrange插值多项式的Marcinkiewicz-Zygmund不等式的最小上界似乎以\（n ^ {\ beta} \），对于某些依赖于\（p> 1 \）的\（\ beta> 0 \）。