Converse Sturm-Hurwitz-Kellogg theorem and related results

We prove that if V ⁿ is a Chebyshev system on the circle and f is a continuous real-valued function with at least n + 1 sign changes then there exists an orientation preserving diffeomorphism of S ¹ that takes f to a function L ²-orthogonal to V. We also prove that if f is a function on the real projective line with at least four sign changes then there exists an orientation preserving diffeomorphism of ℝℙ¹ that takes f to the Schwarzian derivative of a function on ℝℙ¹. We show that the space of piecewise constant functions on an interval with values ± 1 and at most n + 1 intervals of constant sign is homeomorphic to n-dimensional sphere.