We show that if rⁿ (hA),nh = t, is an A-acceptable rational approximation of a strongly continuous semigroup etA 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 etA v as h → 0 under various assumptions on r and v.
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Brenner et al. (1979) studied this question.
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