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...ne to these applications but start very generally. (Of course, one could drop continuity from the assumptions on S towards weaker assumptions, but this is of secondary importance.) Theorem 2.3.4 from =-=[1]-=- states that (1.1) holds in an “integrated form”. From this one can then derive the standard result on semigroups (strong convergence on the domain of the generator). In the paper [3] Driouich and El-...

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... on X and a = 1. Then (1 ∗ S)(t) is convolution with χ[0,t] and (LS)(z) is convolution with e−z, as is easily seen. But the L(X)-norm of the operator “convolution with f” equals the L1-norm of f (see =-=[5]-=- for the easy proof). THE COMPLEX INVERSION FORMULA REVISITED 5 3. Semigroups In this section we apply the results of the previous section to C0-semigroups. Although it is a special case of the situat...

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...hat X has the UMD property the convergence is strong on all of X. This was subsequently generalised from semigroups to solution families for scalar-type Volterra equations by Cioranescu and Lizama in =-=[2]-=-. The aim of the present paper is to present new and much shorter proofs of these results, eventually even generalising them. Our approach uses some elementary Fourier analysis and has the advantage t...

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...eorem 2.3.4 from [1] states that (1.1) holds in an “integrated form”. From this one can then derive the standard result on semigroups (strong convergence on the domain of the generator). In the paper =-=[3]-=- Driouich and El-Mennaoui showed that in case that X has the UMD property the convergence is strong on all of X. This was subsequently generalised from semigroups to solution families for scalar-type ...