2021/11/22 by Sahasranand, K. R. · 1 citation
#15B05 #47A30 #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.2111.11389
A recent paper computed the induced p-norm of a special class of circulant matrices A(n,a,b) ∈ ℝn × n, with the diagonal entries equal to a ∈ ℝ 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. We obtain an exact expression for ‖A‖p, 1 ≤ p ≤ ∞, where A = A(n,a,b), a ≥ 0 and for ‖A‖2 where A = A(n,-a,b), a ≥ 0; for the other p-norms of A(n,-a,b), 2 < p < ∞, we provide upper and lower bounds.