# On an integral transform by R. S. Phillips

Open Mathematics (2010)

- Volume: 8, Issue: 1, page 98-113
- ISSN: 2391-5455

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topSten Bjon. "On an integral transform by R. S. Phillips." Open Mathematics 8.1 (2010): 98-113. <http://eudml.org/doc/269503>.

@article{StenBjon2010,

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 \[ f(t) = \lim \_\{h \rightarrow 0^ + \} \tilde\{f\}\_h (t) \]
, 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.},

author = {Sten Bjon},

journal = {Open Mathematics},

keywords = {One-parameter semigroups; Yosida approximation; Laplace transform; one-parameter semigroups},

language = {eng},

number = {1},

pages = {98-113},

title = {On an integral transform by R. S. Phillips},

url = {http://eudml.org/doc/269503},

volume = {8},

year = {2010},

}

TY - JOUR

AU - Sten Bjon

TI - On an integral transform by R. S. Phillips

JO - Open Mathematics

PY - 2010

VL - 8

IS - 1

SP - 98

EP - 113

AB - 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 \[ f(t) = \lim _{h \rightarrow 0^ + } \tilde{f}_h (t) \]
, 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.

LA - eng

KW - One-parameter semigroups; Yosida approximation; Laplace transform; one-parameter semigroups

UR - http://eudml.org/doc/269503

ER -

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