Converse Sturm-Hurwitz-Kellogg theorem and related results (Q942966)

This article proves a generalization of the converse four-vertex theorem and discusses related results. The Sturm-Hurwitz-Kellogg theorem states that any smooth function on \(S^1\) being orthogonal to a Chebyshev system of dimension \(n\) has at least \(n+1\) sign changes. Applying this to the curvature radius of an oval curve for functions \(\{1,\cos \alpha, \sin\alpha\}\) implies the four vertex theorem. Now the converse to Sturm-Hurwitz-Kellogg theorem says that if a smooth function on \(S^1\) has at least \(n+1\) sign changes, and let \(V\) be a Chebyshev system of dimension \(n\) on \(S^1\), then there exists a diffeomorphism \(\phi\) such that \(f\circ\phi\) is orthogonal to \(V\).