Suppose I have a smooth time series $C(t)$ defined on the interval $t=[0,T]$, from which I extract the sub-series $c=\{x_1,\cdots,x_N\}$ of $N$ entries, where $x_i=C(i*T/N)$. Naturally, the number $N$ defines the sampling frequency, and in the limit of $N\to\infty$ we recover the full signal with infinite detail. I can turn this series $c$ into a Hankel matrix
$H_c=\begin{pmatrix}x_1&x_2& \cdots& x_N\\ x_2&x_3&\cdots&x_1\\ \vdots&\vdots&\ddots&\vdots\\ x_N&x_1&\cdots&x_{N-1}\end{pmatrix}$.
I need to perform a singular value decomposition (SVD) on this Hankel matrix for different values of the sampling frequency, usually powers of 2. My question is, how do the singular values change when you go from $N=16$ to $N=32$ for example? Does there exist a clear-cut relation dependent on $C(t)$? Or do there exist lower/upper bounds on the new singular values $\sigma_k$?
I have attempted some proofs but I quickly found out I do not have the appropriate tools in my toolbox to tackle this problem fully. I also did some numerical simulations of easy series, like $c=\{0,\frac{1}{4},\frac{1}{2},\frac{3}{4}\}\to c=\{0,\frac{1}{8},\frac{1}{4},\frac{3}{8},\frac{1}{2},\frac{5}{8},\frac{3}{4},\frac{7}{8}\}$ to see if there was any correlation between the singular values but failed to find a consistent pattern.