On an integral transform by R. S. Phillips

Sten Bjon

Open Mathematics (2010)

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

Abstract

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The properties of a transformation f 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 ( f ˜ h ) k ˜ = f ˜ h + k for certain complex h and k, and that f ( t ) = lim h 0 + 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.

How to cite

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Sten 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 -

References

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