Hostname: page-component-76fb5796d-22dnz Total loading time: 0 Render date: 2024-04-30T03:46:18.426Z Has data issue: false hasContentIssue false
  • English
  • Français

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, gthorbergsson@mi.uni-koeln.de
Masaaki Umehara
Affiliation:
Department of Mathematics, Graduate School of Science, Osaka University, Osaka, 560-0043, Japan, umehara@math.wani.osaka-u.ac.jp
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

References

[Ar1] Arnold, V., Equations différentielles ordinaires, Editons Mir, Moscou, 1974.Google Scholar
[Ar2] Arnold, V., Remarks on the extatic points of plane curves, The Gelfand Mathematical Seminars, 1993–1995, Birkhäuser, Boston (1996), pp. 11–22.Google Scholar
[Ba] Barner, M., Uber die Mindestanzahl stationärer Schmiegebenen bei geschlossenen strengkonvexen Raumkurven, Abh. Math. Sem. Univ. Hamburg, 20 (1956), 196–215.Google Scholar
[Bl] Blaschke, W., Vorlesungen Über Differentialgeometrie II, Die Grundlehren der Mathematischen Wissenschaften, Band 7, Springer-Verlag, Berlin, 1923.Google Scholar
[Bo] Bose, R. C., On the number of circles of curvature perfectly enclosing or perfectly enclosed by a closed convex oval, Math. Z., 35 (1932), 16–24.Google Scholar
[Co] Coppel, W. A., Disconjugacy, Lecture Notes in Math. 220, Springer-Verlag, Berlin etc., 1971.Google Scholar
[GMO] Guieu, L., Mourre, E. and Ovsienko, V. Yu., Theorem on six vertices of a plane curve via Sturm theory, The Arnold-Gelfand Mathematical Seminars, Birkhäuser, Boston (1997), pp. 257–266.Google Scholar
[Ha] Haupt, O., Verallgemeinerung eines Satzes von R. C. Bose Über die Anzahl der Schmiegkreise eines Ovals, die vom Oval umschlossen werden oder das Oval umschließen, J. Reine Angew. Math., 239/240 (1969), 339–352.Google Scholar
[HK] Haupt, O. and Künneth, H., Geometrische Ordnungen, Die Grundlehren der Mathematischen Wissenschaften, Band 133, Springer-Verlag, Berlin, New York, 1967.Google Scholar
[Hy] Hayashi, T., Some geometrical applications of Fourier series, Rend. Circ. Mat. Palermo, 50 (1926), 96–102.Google Scholar
[Ja] Jackson, S. B., Vertices of plane curves, Bull. Amer. Math. Soc., 50 (1944), 564–578.Google Scholar
[Kn] Kneser, H., Neuer Beweis des Vierscheitelsatzes, Christiaan Huygens, 2 (1922/23), 315–318.Google Scholar
[Mu1] Mukhopadhyaya, S., New methods in the geometry of a plane arc, I, Bull. Calcutta Math. Soc., 1 (1909), 31–37. Also in Collected geometrical papers, vol. I, Calcutta University Press, Calcutta, 1929, pp. 13–20.Google Scholar
[Mu2] Mukhopadhyaya, S., Sur les nouvelles méthodes de géometrie, C. R. Séance Soc. Math. France, année 1933 (1934), 41–45.Google Scholar
[Nö] Nöbeling, G., Uber die Anzahl der ordnungsgeometrischen Scheitel von Kurven II, Geometriae Dedicata, 31 (1989), 137–149.Google Scholar
[OT] Ovsienko, V. and Tabachnikov, S., Sturm theory, Ghys theorem on zeros of the Schwarzian derivative and flattening of Legendrian curves, Selecta Math. (New Series), 2 (1996), 297–307.Google Scholar
[Ta] Tabachnikov, S., Parametrized plane curves, Minkowski caustics, Minkowski vertices and conservative line fields, Enseign. Math. (2), 43 (1997), 3–26.Google Scholar
[TU1] Thorbergsson, G. and Umehara, M., A unified approach to the four vertex theorems, II, Differential and symplectic topology of knots and curves, American Math. Soc. Transl. (Series 2) 190, Amer. Math. Soc., Providence, R.I. (1999), pp. 229–252.Google Scholar
[TU2] Thorbergsson, G. and Umehara, M., Sectactic points on a simple closed curve, Nagoya Math. J., 167 (2002), 55–94.Google Scholar
[Um] Umehara, M., A unified approach to the four vertex theorems, I, Differential and symplectic topology of knots and curves, American Math. Soc. Transl. (Series 2) 190, Amer. Math. Soc., Providence, R.I. (1999), pp. 185–228.Google Scholar