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

top
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.

How to cite

top

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

top
  1. [1] E. Borel, Sur quelques points de la théorie des fonctions, Ann. Sci. Ecole Norm. Sup. 12 (1895), 9-55. Zbl26.0429.03
  2. [2] J. C. Canille, Desenvolvimento asintotico e introduç ao as cálculo diferential resurgente, 17 Colóquio Brasileiro de Matemática, IMPA, 1989. 
  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). 
  6. [6] L. Hörmander, The Analysis of Linear Partial Differential Operators, I, Springer, Berlin, 1983. Zbl0521.35001
  7. [7] S. Mandelbrojt, Séries Adhérentes, Régularisation des Suites, Applications, Collection de Monographies sur la Théorie des Fonctions, Gauthier-Villars, Paris, 1952. 
  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. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.