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On the Yosida approximation and the Widder-Arendt representation theorem. (English) Zbl 0876.44001

For an infinitely differentiable function \(f(\lambda)\) with values in a Banach space with \(|f^{(n)}(\lambda)|\leq Mn!(\lambda- \omega)^{-n-1}\) for all \(n\geq 0\), \(\lambda>\omega\), and a fixed real \(\omega\), the Yosida approximation is defined as \(g_\mu(t)= e^{-\mu t}t\mu^2 \sum^\infty_{n=0} (-t\mu^2)^n f^{(n)}(\mu)/ (n!(n+1)!)\). It is shown that \(u(t)= \lim_{\mu\to\infty} \int_0^t g_\mu(s)ds\) satisfies \(f(\lambda)=\lambda \int_0^\infty e^{-\lambda t}u(t)dt\) for \(\lambda>\omega\). The result can be used in the theory of continuous semigroups.
Reviewer: L.Berg (Rostock)

MSC:

44A10 Laplace transform
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
47D03 Groups and semigroups of linear operators
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