Uniform l1 behaviour for time discretization of a Volterra equation with completely monotonic kernel: I. stability

This paper is the first of two papers on the time discretization of the equation ut + ∫t0 β (t — s) Au (s) ds = 0, t > 0, u (0) = u0, where A is a self‐adjoint densely defined linear operator on a Hilbert space H with a complete eigensystem {λm, φm}∞m = 1, and β (t) is completely monotonic and locally integrable, but not constant. The equation is discretized in time using first‐order differences in combination with order‐one convolution quadrature. The stability properties of the time discretization are derived in the l1t (0, ∞; H) norm.