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

Converse Sturm-Hurwitz-Kellogg theorem and related results