On the Inverse Laplace-stieltjes Transform of A-stable Rational Functions

Let r be an A-stable rational approximation of the exponential function of order q ≥ 1 and let t > 0. It is shown that the inverse LaplaceStieltjes transforms αn : s→ αn∗(ns t ) of rn(z) := r n( tz n ) converge in Lp(R+) to the Heaviside function Ht with a rate of t1/pn−1/2p(ln(n+1))1−1/p. Moreover, for 0 ≤ k ≤ q, the k-th antiderivatives of αn converge in Lp(R+) to the k-th antiderivative of the Heaviside function with a speed that increases with k. In particular, the q-th antiderivatives of αn converge in L1(R+) to the q-th antiderivative of the Heaviside function Ht with the optimal rate of t( t n )q . In addition to the Lp-estimates, bounds on the total variation and supremum norms of αn are given. Via the Hille-Phillips functional calulus for operator semigroups, the results have immediate applications to the error analysis of rational time discretization methods for evolution equations.