A global theory of flexes of periodic functions

Abstract

For a real valued periodic smooth function $u$ on $\R$, $n \ge 0$, one defines the osculating polynomial $\varphi_{s}$ (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 $\varphi_{s} \ge u$ (resp.\ $\varphi_{s} \le u$) and the preimage $(\varphi-u)^{-1}(0)$ is connected. We prove that any smooth periodic function $u$ has 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.