周期函数弯曲的整体理论
A global theory of flexes of periodic functions
摘要：
对于R上的实值周期光滑函数u，$n\ge 0$，定义在点$s\in R$处的密切多项式$ϕ_s$（2n+1阶）为唯一的n次三角多项式，其值和一阶2n阶导数在s处与u在s处的值和一阶2n阶导数一致.我们可以说点s是一个干净的最大弯曲（相应地）。当且仅当$S^1$（分别为$ϕ_s\le u$）和原像$(ϕ-u)^{-1}(0)$连接。证明了任何光滑的周期函数u至少有n+1个2n+1阶的干净的最大弯曲和至少有n+1个2n+1阶的干净的最小弯曲。这一断言显然让人想起莫尔斯理论，并推广了经典的凸平面曲线四点定理。
英文摘要：
For a real valued periodic smooth function u on R, $n\ge 0$, one defines the osculating polynomial $ϕ_s$ (of order 2n+1) at a point $s\in R$ to be the unique trigonometric polynomial of degree n, whose value and first 2n derivatives at s coincide with those of u at s. We will say that a point s is a clean maximal flex (resp. clean minimal flex) of the function u on $S^1$ if and only if $ϕ_s\ge u$ (resp. $ϕ_s\le u$) and the preimage $(ϕ-u)^{-1}(0)$ is connected. We prove that any smooth periodic function u has at least n+1 clean maximal flexes of order 2n+1 and at least n+1 clean minimal flexes of order 2n+1. The assertion is clearly reminiscent of Morse theory and generalizes the classical four vertex theorem for convex plane curves.
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