A Note on the Diagonalization of the Discrete Fourier Transform

Following the approach developed by S. Gurevich and R. Hadani, an analytical formula of the canonical basis of the DFT is given for the case $N=p$ where $p$ is a prime number and $p\equiv 1$ (mod 4).

💡 Research Summary

The paper revisits the diagonalization problem of the discrete Fourier transform (DFT) in the special case where the transform size $N$ is a prime $p$ satisfying $p\equiv1\pmod4$. Building on the representation‑theoretic framework introduced by S. Gurevich and R. Hadani, the author derives an explicit analytical formula for a canonical eigenbasis of the DFT. The core of the construction relies on the finite Heisenberg group $H(\mathbb{F}_p)$, its Weil representation $\rho$ of $SL_2(\mathbb{F}p)$, and the arithmetic properties of quadratic Gauss sums. Because $p\equiv1\pmod4$, the quadratic Gauss sum $G(\chi)=\sum{x\in\mathbb{F}_p}\chi(x)e^{2\pi i x/p}$ is real and equals $\sqrt{p}$ (up to sign). This reality allows the author to define a family of “chirp” vectors
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