A global theory of flexes of periodic functions
| dc.creator | Thorbergsson, Gudlaugur | |
| dc.creator | Umehara, Masaaki | |
| dc.date | 2001-06-12 | |
| dc.date.accessioned | 2026-07-07T04:42:06Z | |
| dc.date.available | 2026-07-07T04:42:06Z | |
| dc.description | 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. | |
| dc.description | 39 pages, 6 figures | |
| dc.identifier | https://arxiv.org/abs/math/0106088 | |
| dc.identifier | http://arxiv.org/abs/math/0106088 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/61632 | |
| dc.subject | Differential Geometry | |
| dc.subject | 51L15, 53C75 | |
| dc.title | A global theory of flexes of periodic functions | |
| dc.type | text |