A geometric characterization of positive definite sequences and of the Fourier transform

If every positive-semidefinite normalized sequence of p autocorrelation coefficients is represented as a point in p-dimensional real space, then the set of such points forms a convex region: the positive semidefinite region. It is shown that the positive semidefinite region can be generated completely as the convex hull of a finite-length, one-dimensional curve that lies on the surface of the region. The curve is specified, several of its properties are given, and it is shown that its length is on the order of p^3/2. The curve represents geometrically the kernel of the Fourier transform; computing the inverse Fourier transform of the spectrum then corresponds to taking the convex linear combination of points on this curve. It is shown that the surface of the positive semidefinite region can then be characterized by a set of polytopes with [p/2]+1 or fewer vertices