逆Sturm-Hurwitz-Kellogg定理及相关结果
Converse Sturm-Hurwitz-Kellogg theorem and related results
摘要：
经典的Sturm-Hurwitz-Kellogg定理断言一个函数，正交于一个圆上的n维切比雪夫系统，至少有n+1个符号变化。我们证明了相反的情况：给定圆上的n维Chebyshev系统和至少具有n+1个符号变化的函数，存在圆的方向保持同构，其将该函数带到与Chebyshev系统正交的函数。我们还证明了，如果一个功能的真正的投影线上有至少四个符号的变化，那么存在一个方向保持的投影线，该函数的Schwarzian导数的一些功能的仿射。这些结果推广了H. Gluck和B. Dahlberg：圆上至少有两个局部极大值和两个局部极小值的函数是闭平面曲线的曲率。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
