# Characterizing the singular values of a matrix with structure

Suppose we have a function from $\mathbb{R}^2\to\mathbb{C}$, $$f(x,y) = e^{\imath\pi x g(y)}$$ where $g(y)$ is periodic in $y\in[-T, T),\ T<\infty$ (e.g., a sinusoid) and $0\leq x < \infty$

Now let $\mathbf{F}\in\mathbb{C}^{n\times m},\ m>n$ be a matrix whose elements $F_{ij}$ are given by $F_{ij} = f(X_i,Y_j)$ where

$$X=\lbrace{0,\ldots,n-1\rbrace}$$

$$Y=\lbrace{-K,-K+\Delta K,\ldots,K\rbrace}, 0< K \le T$$

$$\Delta K=\frac{2K}{m-1}$$

Is it possible to characterize the singular values of $\mathbf{F}$ or the eigenvalues of $\mathbf{FF}^\dagger$ depending on the value $K$ and $m$? For the purposes of this question, assume $g = \sin$.

Some MATLAB code to play with:

f=@(x,y)exp(1i*pi*x*sin(y));
n=50;m=100;K=pi/4;
X=0:n-1;Y=linspace(-K,K,m);
F=f(X',Y);
plot(svd(F))


Looking at the singular values and playing with different $K$ and $m$, my observations are that the choice of $m$ doesn't matter as much as the choice of $K$, with which the singular values show variation. There is certainly structure here, but I'm stuck with trying to formalize the solution.

Any suggestions/pointers to relevant concepts/literature would be appreciated.

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