  We consider half-infinite triangular Toeplitz matrices with slow decay of the
elements and prove under a monotonicity condition that elements of the inverse
matrix, as well as elements of the fundamental matrix, decay to zero. We also
provide a quantitative description of the decay of the fundamental matrix in
terms of p-norms. Finally, we prove that for matrices with slow log-convex
decay the inverse matrix has fast decay, i.e. is bounded. The results are
compared with the classical results of Jaffard and Veccio and illustrated by
numerical example.
