From here, we know the BCCB (block circulant with circulant blocks, a 2 level circulant matrix) can be diagonalized by the Fourier basis. The references therein actually tell us this is true in general for all levels (2 level, 3 level, etc.).
So far so good.
Question
Is the reverse true? Meaning given any diagonal matrix, can I apply the Fourier transform and inverse Fourier transform and get a multilevel circulant matrix?
Specific case of question I am concerned with
Concretely I care about a 3 level circulant matrix (let's say the levels have dimension $n_1, n_2, n_3$ so the total matrix is a square matrix of size $n_1\times n_2\times n_3$ by $n_1\times n_2\times n_3$).
Step 1: Generate an arbitrary vector of length $n_1\times n_2\times n_3$. Then I reshape it into a 3d matrix of size $(n_1, n_2, n_3)$, call it $M$.
Step 2: Let $F$ be the 3D discrete Fourier transform. Then I calculate $F^* \cdot M \cdot F$, call it $A$, where $\cdot$ is matrix multiplication.
Is $A$ a 3 level circulant matrix?