  The classical Hurwitz theorem says that if n first "harmonics" (2n + 1
Fourier coefficients) of a continuous function f(x) on the unit circle are
zero, then f(x) changes sign at least 2n + 1 times. We show that similar facts
and its converse hold for any function that are orthogonal to a Chebyshev
system. These theorems can be extended for convex curves in d-dimensional
Euclidean space. Namely, if a function on a curve is orthogonal to the space of
n-degree polynomials, then the function has at least nd + 1 zeros. This bound
is sharp and is attained for curves on which the space of polynomials forms
classical polynomial and trigonometric Chebyshev systems. We can regard the
theorem of zeros as a generalization of the four-vertex theorem. There exists a
discrete analog of the theorem of zeros for convex polygonal lines which yields
a discrete version of the four-vertex theorem.
