论文标题
无杰克逊型估计分段$ q $ -Sonotone，$ Q \ ge3 $，三角近似
No Jackson-type estimates for piecewise $q$-monotone, $q\ge3$, trigonometric approximation
论文作者
论文摘要
我们说c [a，b] $ in c $ q $ - 单酮，$ q \ ge3 $，如果$ f \ in c^{q-2}（a，b）$和$ f^{（q-2）} $是$（a，b）$的convex $。令$ f $是连续的，$2π$ - 周期性，并以$ [-π，π] $有限地更改其$ Q $ - 单调性。我们有兴趣通过三角多项式估计与之共同Q $ - 单调的$ f $的近似程度，即，三角多项式元素完全改变其$ q $单调性，完全改变了$ f $的位置。这样的杰克逊类型估计值对分段单调（$ q = 1 $）和分段凸（q = 2）近似有效。但是，我们证明，当$ q \ ge3 $时，对于共同的Q $ - 单子酮近似，通常没有此类估计值有效。
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 $q\ge3$.
