The p-norm of circulant matrices via Fourier analysis

A recent work derived expressions for the induced p-norm of a special class of circulant matrices A(n, a, b) is an element of R-nxn with the diagonal entries equal to a is an element of N and the off-diagonal entries equal to b >= 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. The results comprise an exact expression for parallel to A parallel to(p), 1 <= p <= infinity , where A = A(n, a, b), a >= 0 and for parallel to A parallel to(2) where A = A(n, -a, b), a >= 0; for the other p-norms of A(n, -a, b), 2 < p < infinity, upper and lower bounds are derived.