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We show that if rⁿ (hA), nh = t, is an A-acceptable rational approximation of a strongly continuous semigroup e^tA on a Banach space, then for t bounded, \| rⁿ (hA) \| Cn^{1/2}, with certain improvements under additional hypotheses on r. We also discuss the convergence of rⁿ (hA) v to e^tA v as h 0 under various assumptions on r and v.
Brenner et al. (Wed,) studied this question.