论文标题
在Marcinkiewicz-Zygmund不平等和$ a_p $ - $ l $ shape arcs
On Marcinkiewicz-Zygmund inequalities and $A_p$-weights for $L$-shape arcs
论文作者
论文摘要
令$γ$为$ l $形状弧，由2行段组成，与复杂$ z $ -plane $ \ cc $中的$π$相遇的角度不同。本文是为了研究$ \ {z__ {n，k} =ψ（e^{i（2kπ+θ）/（n+ 1）}）$ $ \ {z__ {n，k} =ψ（z__ {n，k} =ψ（对于$θ$）的任何选择。在这方面，我们回想起[-1，1]间隔，fejér点$ \ {z_ {z_ {n，k} =ψ^**（e^{i（2k+1）π/（n+1）} \} $与Chebyshev的点同一致，并且最常用的点是Nagrane nodols nodol and polynol。另一方面，数值实验表明，对于典型的开放$ l $ -l $ shape $γ$，lebesgue常数倾向于以$ o（（log（log（n））^2）$为$ o（（log（n））$），作为多项式$ n $，而$ a_ {p} $ - a_ {p} $ a g}不要从[-1，1]转移到真正的$ L $形状弧。进一步的数值实验还表明，Marcinkiewicz-Zygmund的最小上限对于规范的Lagrange interpolation interpolation多项式为$ \ {z_ {z_ {n，k} \} $似乎以$ n^β$的速率增长，对于$ n^β$，对于某些$ bef> 0 $ phe $ phy $ p> $ p> 1 $ p> 1 $ p> 1 $ p> 1 $ p> 1 $ p> 1 $ p。
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 $p >1$.
