Functions of Difference Matrices Are Toeplitz Plus Hankel

When the heat equation and wave equation are approximated by ut = −Ku and utt = −Ku (discrete in space), the solution operators involve e−Kt, √K, cos(√Kt), and sinc( √ 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-shiftinvariant) 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.