Let f be a real continuous 2Pi (= 2 x 3.14...) -periodic function with Fourier series having null all the harmonic components of order < N.

The Sturm-Hurwitz Theorem states that f has at least 2N distinct zeros in [0, 2Pi) interval.

From original (1836) Sturm's proof, some nice proofs have been proposed, some simple and short (see attached paper). What are the more simple and useful proofs?

Gianluca