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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  1. [1] Beattie R., Butzmann H.-P., Convergence structures and applications to functional analysis, Kluwer Academic Publishers, Dordrecht, 2002 Zbl1246.46003
  2. [2] Binz E., Keller H.H., Funktionenräume in der Kategorie der Limesräume, Ann. Acad. Sci. Fenn. Ser. A I, 1966, 383, 1–21 Zbl0158.19903
  3. [3] Bjon S., Lindström M., A general approach to infinite-dimensional holomorphy, Monatsh. Math., 1986, 101, 11–26 http://dx.doi.org/10.1007/BF01326843 Zbl0579.46031
  4. [4] Bjon S., Einbettbarkeit in den Bidualraum und Darstellbarkeit als projektiver Limes in Kategorien von Limesvektorräumen, Math. Nachr., 1979, 97, 103–116 http://dx.doi.org/10.1002/mana.19790930108 Zbl0441.46003
  5. [5] Bjon S., On an exponential law for spaces of holomorphic mappings, Math. Nachr., 1987, 131, 201–204 http://dx.doi.org/10.1002/mana.19871310118 Zbl0635.54007
  6. [6] Bjon S., Differentiation under the integral sign and holomorphy, Math. Scand., 1987, 60, 77–95 Zbl0613.46043
  7. [7] Bjon S., One-parameter semigroups of operators on L c-embedded spaces, Semigroup Forum, 1994, 49, 369–380 http://dx.doi.org/10.1007/BF02573497 
  8. [8] Bobrowski A., On the Yosida approximation and the Widder-Arendt representation theorem, Studia Math., 1997, 124, 281–290 Zbl0876.44001
  9. [9] Bobrowski A., Inversion of the Laplace transform and generation of Abel summable semigroups, J. Funct. Anal., 2001, 186, 1–24 http://dx.doi.org/10.1006/jfan.2001.3782 
  10. [10] Frölicher A., Bucher W., Calculus in vector spaces without norm, Lecture Notes in Mathematics, 30, Springer-Verlag, Berlin, 1966 Zbl0156.38303
  11. [11] Hille E., Phillips R.S., Functional analysis and semi-groups, Amer. Math. Soc. Colloq. Publ., Vol. XXXI, Amer. Math. Soc., Providence, 1957 Zbl0078.10004
  12. [12] Ladyzhenskaya O., Attractors for semigroups and evolution equations, Cambridge University Press, Cambridge, 1991 Zbl0755.47049
  13. [13] Müller B., L c- und c-einbettbare Limesräume, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser., 1978, 5, 509–526 
  14. [14] Phillips R.S., An inversion formula for Laplace transforms and semi-groups of linear operators, Annals of Math., 1954, 59, 325–356 http://dx.doi.org/10.2307/1969697 Zbl0059.10704
  15. [15] Prudnikov A.P., Brychkov Yu.A., Marichev O.I., Integrals and series, Vol. 4: Direct Laplace transforms, Gordon and Breach Sci. Publ., New York, 1992 Zbl0786.44003

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