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A global theory of flexes of periodic functions

Published online by Cambridge University Press:  22 January 2016

Gudlaugur Thorbergsson  and
Masaaki Umehara
Gudlaugur Thorbergsson
Affiliation:
Mathematisches Institut, Universität zu Köln, Weyertal 86-90, 50931 Köln, Germany, [email protected]
Masaaki Umehara
Affiliation:
Department of Mathematics, Graduate School of Science, Osaka University, Osaka, 560-0043, Japan, [email protected]
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Abstract

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For a real valued periodic smooth function u on R, n ≥ 0, one defines the osculating polynomial φs (of order 2n + 1) at a point s ∈ 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 S1 if and only if φs ≥ u (resp. φs ≤ 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.

Keywords

51L1553C7553A15
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 173 , 2004 , pp. 85 - 138
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2004

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