论文标题
凸出式多项式近似的插值估计值
Interpolatory pointwise estimates for convex polynomial approximation
论文作者
论文摘要
本文涉及凸代数多项式间隔的平滑凸功能$ f $的近似值，这些凸式多项式在此间隔的端点处插入$ f $。我们称此类估计值为“插值”。我们主要定理的一项重要推论是从Δ^{（2）} $近似$ f \ y的结果，凸函数集，从 $ w^r $，$ [-1,1] $上的功能空间，$ f^{（r-1）} $绝对连续，$ \ | f^{（r）} \ | _ {\ iffty}：= = ess \ sup_ {sup_ {x \ in [-1,1]}} | f^{x | f^{（r） <\ infty $： 对于任何$ f \ in w^r \capδ{（2）} $，$ r \ in {\ mathbb n} $，存在一个数字$ {\ mathcal n} = {\ mathcal n}（f，f，r）$在$δ^{（2）} $中，因此\ [ \ left \ | \ frac {f-p_n} {φ^r} \ right \ | _ {\ infty} \ leq \ frac \ frac {c（r）} {n^r} \ left \ | f^{（r）} \ right \ | _ {\ infty}，\]其中$φ（x）：= \ sqrt {1-x^2} $。 对于$ r = 1 $和$ r = 2 $，上述结果以$ {\ mathcal n} = 1 $保留，并且众所周知。对于$ r \ ge 3 $，总体上是不正确的，$ {\ Mathcal n} $独立于$ f $。
This paper deals with approximation of smooth convex functions $f$ on an interval by convex algebraic polynomials which interpolate $f$ at the endpoints of this interval. We call such estimates "interpolatory". One important corollary of our main theorem is the following result on approximation of $f\in Δ^{(2)}$, the set of convex functions, from $W^r$, the space of functions on $[-1,1]$ for which $f^{(r-1)}$ is absolutely continuous and $\|f^{(r)}\|_{\infty} := ess\,sup_{x\in[-1,1]} |f^{(r)}(x)| < \infty$: For any $f\in W^r \capΔ^{(2)}$, $r\in {\mathbb N}$, there exists a number ${\mathcal N}={\mathcal N}(f,r)$, such that for every $n\ge {\mathcal N}$, there is an algebraic polynomial of degree $\le n$ which is in $Δ^{(2)}$ and such that \[ \left\| \frac{f-P_n}{φ^r} \right\|_{\infty} \leq \frac{c(r)}{n^r} \left\| f^{(r)}\right\|_{\infty} , \] where $φ(x):= \sqrt{1-x^2}$. For $r=1$ and $r=2$, the above result holds with ${\mathcal N}=1$ and is well known. For $r\ge 3$, it is not true, in general, with ${\mathcal N}$ independent of $f$.
