论文标题
周期函数的几乎共酮近似程度
Degree of nearly comonotone approximation of periodic functions
论文作者
论文摘要
令$2π$ - 周期函数$ f \ in \ bbb c $在该期间的有限点$ y_i $中更改其单调性。与之共孔酮的三角分配多项式对此$ f $的近似程度，即，$ f $ do的$ y_i $ do的点恰好更改其单调性，受$ω_2（f，π/n）$的限制（根据这些$ y_i $的位置，$ω_2（f，π/n）$）。最近，我们证明，在点$ y_i $（所谓的接近共酮近似值）上，以与$π/n $成正比的间隔放松共鸣性要求，使多项式可以达到近似值$ω_3$。通过构建反例，我们在这里表明，即使放宽了共态性对多项式的要求，该措施接近$ 0 $（无论较慢或多快），$ω_4$无法达到。
Let a $2π$-periodic function $f\in\Bbb C$ changes its monotonicity at a finitely even number of points $y_i$ of the period. The degree of approximation of this $f$ by trigonometric polynomials which are comonotone with it, i.e. that change their monotonicity exactly at the points $y_i$ where $f$ does, is restricted by $ω_2(f,π/n)$ (with a constant depending on the location of these $y_i$). Recently, we proved that relaxing the comonotonicity requirement in intervals of length proportional to $π/n$ about the points $y_i$ (so called nearly comonotone approximation) allows the polynomials to achieve the approximation rate of $ω_3$. By constructing a counterexample, we show here that even with the relaxation of the requirement of comonotonicity for the polynomials on sets with measures approaching $0$ (no matter how slowly or how fast) $ω_4$ is not reachable.
