On the decay of elements of inverse triangular Toeplitz matrix

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.