分段$q$ -单调、$q\ge3$、三角逼近的非Jackson型估计
No Jackson-type estimates for piecewise $q$-monotone, $q\ge3$, trigonometric approximation
摘要：
我们说一个函数$f\in C[a,b]$是$q$ -单调的，$q\ge3$，如果$f\in C^{q-2}(a,b)$和$f^{(q-2)}$在$(a,b)$中是凸的。设$f$是连续的，$2π$是周期的，并且在$[-π,π]$中多次改变它的$q$ -单调性.我们感兴趣的是估计程度的逼近$f$三角多项式是共同的$q$ -单调与它，即三角多项式改变其$q$ -单调性正是在点$f$没有。这种Jackson型估计对于分段单调（$q=1$）和分段凸（q=2）近似是有效的。然而，我们证明，没有这样的估计是有效的，在一般情况下，对共同$q$ -单调逼近，当$q\ge3$。
英文摘要：
We say that a function $f\in C[a,b]$ is $q$-monotone, $q\ge3$, if $f\in C^{q-2}(a,b)$ and $f^{(q-2)}$ is convex in $(a,b)$. Let $f$ be continuous and $2π$-periodic, and change its $q$-monotonicity finitely many times in $[-π,π]$. We are interested in estimating the degree of approximation of $f$ by trigonometric polynomials which are co-$q$-monotone with it, namely, trigonometric polynomials that change their $q$-monotonicity exactly at the points where $f$ does. Such Jackson type estimates are valid for piecewise monotone ($q=1$) and piecewise convex (q=2) approximations. However, we prove, that no such estimates are valid, in general, for co-$q$-monotone approximation, when $q\ge3$.
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