Fourier Series of Polygons

say that such a function f is a polygon if there is a finite subdivision O < to < * * * < tj < tj + 1 * * * < tn-1 < tn to + 2 v of [O, 2v] such that f is affine linear in each subinterval [tj, tj+ 1[ In this case, the image of f is a polygonal line in ¢ with vertices sj = f(tj), and f is a parametrization with constant speed on each side. The set of polygons is obviously a subspace of W(R/2v@). Let us now determine the Fourier series of a polygon f. With the previous notations, we compute the Fourier coefficients Ck( f ) given by