On the trigonometric moment problem in two dimensions

A finite sequence c0,…,ckc0,…,ck of complex numbers is the set of trigonometric moments of a measure dμ=∣Pk(eiθ)∣−2dθdμ=∣Pk(eiθ)∣−2dθ with Pk(z)Pk(z) a polynomial of degree kk, zero-free in |z|≤1|z|≤1, providing only that the Hermitian Toeplitz matrix having c0,…,ckc0,…,ck as its top row is positive-definite. The analogous representation in two dimensions for a rectangular array of numbers requires additional conditions, discovered by J.S. Geronimo and H.J. Woerdeman. We use orthogonal decomposition to give an alternate proof of their important result.