Converse Sturm-Hurwitz-Kellogg theorem and related results

TL;DR

This paper proves a converse to the classical Sturm-Hurwitz-Kellogg theorem, showing that functions with sufficient sign changes can be transformed into orthogonal functions or Schwarzian derivatives via circle diffeomorphisms, extending classical geometric results.

Contribution

It establishes the converse of the Sturm-Hurwitz-Kellogg theorem and related results, linking sign changes to the existence of specific diffeomorphisms and extending classical geometric theorems.

Findings

Functions with enough sign changes can be transformed into orthogonal functions.

Extension of the four vertex theorem to functions on the circle and projective line.

Existence of diffeomorphisms mapping functions to Schwarzian derivatives.

Abstract

The classical Sturm-Hurwitz-Kellogg theorem asserts that a function, orthogonal to an n-dimensional Chebyshev system on a circle, has at least n+1 sign changes. We prove the converse: given an n-dimensional Chebyshev system on a circle and a function with at least n+1 sign changes, there exists an orientation preserving diffeomorphism of the circle that takes this function to a function, orthogonal to the Chebyshev system. We also prove that if a function on the real projective line has at least four sign changes then there exists an orientation preserving diffeomorphism of the projective line that takes this function to the Schwarzian derivative of some function. These results extend the converse four vertex theorem of H. Gluck and B. Dahlberg: a function on a circle with at least two local maxima and two local minima is the curvature of a closed plane curve.