Functions of Difference Matrices Are Toeplitz Plus Hankel

When the heat equation and wave equation are approximated by u(t) = -Ku and u(tt) = -Ku (discrete in space), the solution operators involve e(-Kt), root K, cos(root Kt), and sinc(root Kt). We compute these four matrices and find accurate approximations with a variety of boundary conditions. The second difference matrix K is Toeplitz (shift-invariant) for Dirichlet boundary conditions, but we show why e(-Kt) also has a Hankel (anti-shift-invariant) part. Any symmetric choice of the four corner entries of K leads to Toeplitz plus Hankel in all functions f(K). Overall, this article is based on diagonalizing symmetric matrices, replacing sums by integrals, and computing Fourier coefficients.