The $p$-norm of circulant matrices via Fourier analysis

A recent paper computed the induced $p$-norm of a special class of circulant matrices $A(n,a,b) \in \mathbb{R}^{n \times n}$, with the diagonal entries equal to $a \in \mathbb{R}$ and the off-diagonal entries equal to $b \ge 0$. We provide shorter proofs for all the results therein using Fourier analysis. The key observation is that a circulant matrix is diagonalized by a DFT matrix. We obtain an exact expression for $\|A\|_p, 1 \le p \le \infty$, where $A = A(n,a,b), a \ge 0$ and for $\|A\|_2$ where $A = A(n,-a,b), a \ge 0$; for the other $p$-norms of $A(n,-a,b)$, $2 < p < \infty$, we provide upper and lower bounds.