Extension maps in ultradifferentiable and ultraholomorphic function spaces

Jean Schmets; Manuel Valdivia

Studia Mathematica (2000)

  • Volume: 143, Issue: 3, page 221-250
  • ISSN: 0039-3223

Abstract

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The problem of the existence of extension maps from 0 to ℝ in the setting of the classical ultradifferentiable function spaces has been solved by Petzsche [9] by proving a generalization of the Borel and Mityagin theorems for -spaces. We get a Ritt type improvement, i.e. from 0 to sectors of the Riemann surface of the function log for spaces of ultraholomorphic functions, by first establishing a generalization to some nonclassical ultradifferentiable function spaces.

How to cite

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Schmets, Jean, and Valdivia, Manuel. "Extension maps in ultradifferentiable and ultraholomorphic function spaces." Studia Mathematica 143.3 (2000): 221-250. <http://eudml.org/doc/216817>.

@article{Schmets2000,
abstract = {The problem of the existence of extension maps from 0 to ℝ in the setting of the classical ultradifferentiable function spaces has been solved by Petzsche [9] by proving a generalization of the Borel and Mityagin theorems for $C^\{∞\}$-spaces. We get a Ritt type improvement, i.e. from 0 to sectors of the Riemann surface of the function log for spaces of ultraholomorphic functions, by first establishing a generalization to some nonclassical ultradifferentiable function spaces.},
author = {Schmets, Jean, Valdivia, Manuel},
journal = {Studia Mathematica},
keywords = {extension map; ultradifferentiable function; Roumieu type; Beurling type; Borel theorem; ultraholomorphic functions; (LB)-space; surjectivity of the restriction map; ultradifferentiable function spaces of Beurling and Roumieu types; Fréchet space},
language = {eng},
number = {3},
pages = {221-250},
title = {Extension maps in ultradifferentiable and ultraholomorphic function spaces},
url = {http://eudml.org/doc/216817},
volume = {143},
year = {2000},
}

TY - JOUR
AU - Schmets, Jean
AU - Valdivia, Manuel
TI - Extension maps in ultradifferentiable and ultraholomorphic function spaces
JO - Studia Mathematica
PY - 2000
VL - 143
IS - 3
SP - 221
EP - 250
AB - The problem of the existence of extension maps from 0 to ℝ in the setting of the classical ultradifferentiable function spaces has been solved by Petzsche [9] by proving a generalization of the Borel and Mityagin theorems for $C^{∞}$-spaces. We get a Ritt type improvement, i.e. from 0 to sectors of the Riemann surface of the function log for spaces of ultraholomorphic functions, by first establishing a generalization to some nonclassical ultradifferentiable function spaces.
LA - eng
KW - extension map; ultradifferentiable function; Roumieu type; Beurling type; Borel theorem; ultraholomorphic functions; (LB)-space; surjectivity of the restriction map; ultradifferentiable function spaces of Beurling and Roumieu types; Fréchet space
UR - http://eudml.org/doc/216817
ER -

References

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  3. [3] H. Cartan, Sur les classes de fonctions définies par des inégalités portant sur leurs dérivées successives, Actualités Sci. Indust. 867, Publ. Inst. Math. Univ. Clermont-Ferrand, Hermann, Paris, 1940. Zbl0061.11701
  4. [4] A. Gorny, Contribution à l'étude de fonctions dérivables d'une variable réelle, Acta Math. 71 (1939), 317-358. Zbl65.0216.01
  5. [5] A. Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. 16 (1955). 
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  8. [8] B. Mityagin, Approximate dimension and bases in nuclear spaces, Uspekhi Mat. Nauk 16 (1961), no. 4, 63-132 (in Russian); English transl.: Russian Math. Surveys 16 (1961), 59-127. 
  9. [9] H.-J. Petzsche, On E. Borel's theorem, Math. Ann. 282 (1988), 299-313. Zbl0633.46033
  10. [0] J. F. Ritt, On the derivatives of a function at a point, Ann. of Math. 18 (1916), 18-23. Zbl46.0471.02
  11. [1] J. C. Tougeron, An introduction to the theory of Gevrey expansions and to the Borel-Laplace transform with some applications, Course of 3rd Cycle, Univ. of Toronto. 
  12. [2] G. Valiron, Théorie des fonctions, Masson, Paris, 1966. 

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