Abstract
The properties of a transformation \( f \mapsto \tilde f_h \) by R.S. Phillips, which transforms an exponentially bounded C 0-semigroup of operators T(t) to a Yosida approximation depending on h, are studied. The set of exponentially bounded, continuous functions f: [0, ∞[→ E with values in a sequentially complete L c -embedded space E is closed under the transformation. It is shown that \( (\tilde f_h )\widetilde{_k } = \tilde f_{h + k} \) for certain complex h and k, and that \( f(t) = \lim _{h \to 0^ + } \tilde f_h (t) \), where the limit is uniform in t on compact subsets of the positive real line. If f is Hölder-continuous at 0, then the limit is uniform on compact subsets of the non-negative real line. Inversion formulas for this transformation as well as for the Laplace transformation are derived. Transforms of certain semigroups of non-linear operators on a subset X of an L c -embedded space are studied through the C 0-semigroups, which they define by duality on a space of functions on X.
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Bjon, S. On an integral transform by R. S. Phillips. centr.eur.j.math. 8, 98–113 (2010). https://doi.org/10.2478/s11533-009-0058-8
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DOI: https://doi.org/10.2478/s11533-009-0058-8