凸多项式逼近的插值点态估计
Interpolatory pointwise estimates for convex polynomial approximation
摘要：
本文讨论了区间上光滑凸函数$f$的凸代数多项式逼近问题，该多项式在区间的端点上插值$f$。我们称这种估计为“插值”。我们的主要定理的一个重要推论是下面的结果逼近$f\in Δ^{(2)}$，凸函数的集合，从$W^r$，$f^{(r-1)}$是绝对连续的$[-1,1]$上的函数空间和$\|f^{(r)}\|_{\infty} := ess\,sup_{x\in[-1,1]} |f^{(r)}(x)| < \infty$：对于任何$f\in W^r \capΔ^{(2)}$，$r\in {\mathbb N}$，存在一个数字${\mathcal N}={\mathcal N}(f,r)$，使得对于每个$n\ge {\mathcal N}$，存在次数为$\le n$的代数多项式，其在$Δ^{(2)}$中，并且使得\[ \left\| \frac{f-P_n}{φ^r} \right\|_{\infty} \leq \frac{c(r)}{n^r} \left\| f^{(r)}\right\|_{\infty} , \]其中$φ(x):= \sqrt{1-x^2}$。对于$r=1$和$r=2$，上述结果对于${\mathcal N}=1$成立，并且是公知的。对于$r\ge 3$，一般来说，${\mathcal N}$独立于$f$是不正确的。
英文摘要：
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 $f$.
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