# Extension of distributions and representation by derivatives of continuous functions.

- Volume: 7, Issue: 1, page 31-40
- ISSN: 1120-6330

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topLemoine, Jérôme, and Simon, Jacques. "Extension of distributions and representation by derivatives of continuous functions.." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 7.1 (1996): 31-40. <http://eudml.org/doc/244168>.

@article{Lemoine1996,

abstract = {It is proved that any Banach valued distribution on a bounded set can be extended to all of \( \mathbb\{R\}^\{d\} \) if and only if it is a derivative of a uniformly continuous function. A similar result is given for distributions on an unbounded set. An example shows that this does not extend to Frechet valued distributions. This relies on the fact that a Banach valued distribution is locally a derivative of a uniformly continuous function. For sake of completeness, a global representation of a Banach valued distribution by derivatives of functions with compact supports is given.},

author = {Lemoine, Jérôme, Simon, Jacques},

journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},

keywords = {Distributions; Extension; Vector-values; Representation; Banach valued distribution on a bounded set; Fréchet valued distributions; global representation of a Banach valued distribution by derivatives of functions with compact supports},

language = {eng},

month = {5},

number = {1},

pages = {31-40},

publisher = {Accademia Nazionale dei Lincei},

title = {Extension of distributions and representation by derivatives of continuous functions.},

url = {http://eudml.org/doc/244168},

volume = {7},

year = {1996},

}

TY - JOUR

AU - Lemoine, Jérôme

AU - Simon, Jacques

TI - Extension of distributions and representation by derivatives of continuous functions.

JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

DA - 1996/5//

PB - Accademia Nazionale dei Lincei

VL - 7

IS - 1

SP - 31

EP - 40

AB - It is proved that any Banach valued distribution on a bounded set can be extended to all of \( \mathbb{R}^{d} \) if and only if it is a derivative of a uniformly continuous function. A similar result is given for distributions on an unbounded set. An example shows that this does not extend to Frechet valued distributions. This relies on the fact that a Banach valued distribution is locally a derivative of a uniformly continuous function. For sake of completeness, a global representation of a Banach valued distribution by derivatives of functions with compact supports is given.

LA - eng

KW - Distributions; Extension; Vector-values; Representation; Banach valued distribution on a bounded set; Fréchet valued distributions; global representation of a Banach valued distribution by derivatives of functions with compact supports

UR - http://eudml.org/doc/244168

ER -

## References

top- DUGUNDJI, J., Topology. Allyn and Bacon, Inc., Boston1966. Zbl0397.54003MR193606
- SCHWARTZ, L., Théorie des distributions à valeurs vectorielles. Annales de l'Institut Fourier, tome VII, 1957. Zbl0078.11003
- SCHWARTZ, L., Théorie des distributions. Hermann 1966 (ed. of 1973). Zbl0149.09501MR209834
- SIMON, J., Distributions à valeurs dans un espace séquentiellement complet. Preprint of Laboratoire de Mathématiques Appliquées, Université Blaise Pascal, 1996, to appear.

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