论文标题
通过多项式插值在复杂平面及以外的多项式插值的统一近似顺序
Order of uniform approximation by polynomial interpolation in the complex plane and beyond
论文作者
论文摘要
对于在$ \ mathbb {c} $中的lagrange多项式插值中，众所周知，Chebyshev家族的lebesgue常数$ {\ bf {x}}} _ n： $ [ - 1,1] \ subset \ mathbb {r} $的增长顺序为$ O（log（n））$。在[45]的[45]中，家庭的lebesgue $ {\ bf {\ bf {z^{**} _ n}}}}}}}：= \ {z__ {z_ {n，j}^{**} \} \} \}^{n}^{n} _ { \ Mathbb {C} $。另一方面，在我们最近的工作[15]中，可以观察到，如果由$ l $ -Shape arc $γ_0\ subset \ subset \ subset \ mathbb {c} $取代，由两个线段组成z} _n^{*}：= \ {z_ {z {n，j}^{*} \} \}^{n} _ {n} _ {j = 0} $ on $γ$，以及相应的lebesgue contancter lebesgue常数$ l _ {z}^n}的增长率一些常数$ c> 0 $。 本文的主要目的是3倍：首先，将表明，对于$ l $ -shape arc $γ_0$的特殊情况，由两个线段组成的两个线段组成，其长度相同，与$π/2 $相遇的相同长度，lebesgue constant $ l _ { $ o（log^2（n））$，带有$ \ lim \ sup \ frac {l _ {\ bf {z} _n^{*}}}}}} {log^2（n）} = \ infty $;其次，相应的（修改的）Marcinkiewicz-Zygmund的不平等现象无法持有；第三，适当调整$ {\ bf z} _n^{**}：= \ {z__ {z_ {n，j}^{**}^{**} \}^{n} _ {n} _ {j = 0} $ offejér积分的$γ$的增长利率将被描述z} _n^{**}} $正好为$ o（log^2（n））$。
For Lagrange polynomial interpolation on open arcs $X=γ$ in $\mathbb{C}$, it is well-known that the Lebesgue constant for the family of Chebyshev points ${\bf{x}}_n:=\{x_{n,j}\}^{n}_{j=0}$ on $[-1,1]\subset \mathbb{R}$ has growth order of $O(log(n))$. The same growth order was shown in [45] for the Lebesgue constant of the family ${\bf {z^{**}_n}}:=\{z_{n,j}^{**}\}^{n}_{j=0}$ of some properly adjusted Fejér points on a rectifiable smooth open arc $γ\subset \mathbb{C}$. On the other hand, in our recent work [15], it was observed that if the smooth open arc $γ$ is replaced by an $L$-shape arc $γ_0 \subset \mathbb{C}$ consisting of two line segments, numerical experiments suggest that the Marcinkiewicz-Zygmund inequalities are no longer valid for the family of Fejér points ${\bf z}_n^{*}:=\{z_{n,j}^{*}\}^{n}_{j=0}$ on $γ$, and that the rate of growth for the corresponding Lebesgue constant $L_{\bf {z}^{*}_n}$ is as fast as $c\,log^2(n)$ for some constant $c>0$. The main objective of the present paper is 3-fold: firstly, it will be shown that for the special case of the $L$-shape arc $γ_0$ consisting of two line segments of the same length that meet at the angle of $π/2$, the growth rate of the Lebesgue constant $L_{\bf {z}_n^{*}}$ is at least as fast as $O(Log^2(n))$, with $\lim\sup \frac{L_{\bf {z}_n^{*}}}{log^2(n)} = \infty$; secondly, the corresponding (modified) Marcinkiewicz-Zygmund inequalities fail to hold; and thirdly, a proper adjustment ${\bf z}_n^{**}:=\{z_{n,j}^{**}\}^{n}_{j=0}$ of the Fejér points on $γ$ will be described to assure the growth rate of $L_{{\bf z}_n^{**}}$ to be exactly $O(Log^2(n))$.
