Extension of distributions and representation by derivatives of continuous functions.

Jérôme Lemoine; Jacques Simon

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1996)

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

Abstract

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It is proved that any Banach valued distribution on a bounded set can be extended to all of 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.

How to cite

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Lemoine, 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 -

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