At first the simple case. Let $h_kin mathbb C$ of the form

$$h_k=sum_{j=1}^Mc_j;z_j^kqquadqquad text{for }kinmathbb Z_{>0}.$$

Where $c_jin mathbb C$ are some weights and $z_jin mathbb C$ pairwise distinct with $|z_j|=1$. The normalize condition should not be that important, but maybe this additional property is getting useful for my main question. Now lets define the Hankel matrix $H(s)$ for $sin mathbb N_0$ given by

$$H(s)=(h_{s+m+l})_{m=0,l=0}^{M-1,M-1}=begin{pmatrix}h_{s}&dots&h_{M-1+s}\vdots&&vdots\h_{M-1+s}&dots&h_{2M-2+s}end{pmatrix}.$$

We can now find a decomposition of this matrix by

begin{align*} H(s)&=begin{pmatrix}z_1^{s}&dots&z_M^s\vdots&&vdots&\z_1^{M-1+s}&dots&z_M^{M-1+s}end{pmatrix}

begin{pmatrix} c_1&&\&ddots\&&c_Mend{pmatrix}

begin{pmatrix}1&dots&z_1^{M-1}\vdots&&vdots&\1&dots&z_M^{M-1}end{pmatrix}.end{align*}

We can than look on the 3 different determinants and will see, that $H(s)$ is indeed invertible. Ok so far so good.

Now to my main question. We will now look on $h_k$ defined by

$$h_k=sum_{j=1}^Mphi_j(k);z_j^k.$$

Where $phi_j$ are some complex polynomials of degree $d$ and $z_j$ as before. Again we are looking on the Hankel matrix

$$h(s)=(h_{s+m+l})^{(d+1)cdot(M-1)}_{m,l=0}.$$

So it is the ‘same’ matrix, but now with new coefficients and bigger.

I assume that kind of matrix to be invertible since, I am calculating a lot with them and they was always (even numerical) invertible. Furthermore I found some remarks where they said they should be invertible. How can I show, that this new Hankel matrix is still invertible? Is there an similar decomposition like the simple case before?

This problem is strongly related with Prony’s Method (maybe it will help you).