论文标题
Littlewood多项式的阶梯根和高加ak级功能的极值
Step roots of Littlewood polynomials and the extrema of functions in the Takagi class
论文作者
论文摘要
我们提供了一种新的方法来表征和计算高加台类，尤其是高加族人函数的函数的全球最大化器和最小化器集。后者形成了分形函数的家族$f_α：[0,1] \ to \ mathbb r $由$α\ in（-2,2）$参数化。我们表明，$f_α$在$ [0,1/2] $中具有唯一的最大化器，并且仅当不存在以$α$作为某种类型的root（称为步骤root）的Littlewood多项式。 Our general results lead to explicit and closed-form expressions for the maxima of the Takagi--Landsberg functions with $α\in(-2,1/2]\cup(1,2)$. For $(1/2,1]$, we show that the step roots are dense in that interval. If $α\in (1/2,1]$ is a step root, then the set of maximizers of $f_α$ is an明确地使用Hausdorff Dimension $ 1/（N+1）$，其中$ n $是最小的Littlewood多项式的程度，其$α$是其阶跃根，我们以相同的方式确定所有Thice Is of All Little Worder of All Little Worder of All Little Worder， $ [-2，-1/2] \ cup [1/2,2] $。
We give a new approach to characterizing and computing the set of global maximizers and minimizers of the functions in the Takagi class and, in particular, of the Takagi--Landsberg functions. The latter form a family of fractal functions $f_α:[0,1]\to\mathbb R$ parameterized by $α\in(-2,2)$. We show that $f_α$ has a unique maximizer in $[0,1/2]$ if and only if there does not exist a Littlewood polynomial that has $α$ as a certain type of root, called step root. Our general results lead to explicit and closed-form expressions for the maxima of the Takagi--Landsberg functions with $α\in(-2,1/2]\cup(1,2)$. For $(1/2,1]$, we show that the step roots are dense in that interval. If $α\in (1/2,1]$ is a step root, then the set of maximizers of $f_α$ is an explicitly given perfect set with Hausdorff dimension $1/(n+1)$, where $n$ is the degree of the minimal Littlewood polynomial that has $α$ as its step root. In the same way, we determine explicitly the minima of all Takagi--Landsberg functions. As a corollary, we show that the closure of the set of all real roots of all Littlewood polynomials is equal to $[-2,-1/2]\cup[1/2,2]$.
