Some classes of infinitely differentiable functions

G. S. Balashova

Mathematica Bohemica (1999)

  • Volume: 124, Issue: 2-3, page 167-172
  • ISSN: 0862-7959

Abstract

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For nonquasianalytical Carleman classes conditions on the sequences { M ^ n } and { M n } are investigated which guarantee the existence of a function in C J { M ^ n } such that u(n)(a) = bn,    bnKn+1Mn,    n = 0,1,...,    aJ. Conditions of coincidence of the sequences { M ^ n } and { M n } are analysed. Some still unknown classes of such sequences are pointed out and a construction of the required function is suggested. The connection of this classical problem with the problem of the existence of a function with given trace at the boundary of the domain in a Sobolev space of infinite order is shown.

How to cite

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Balashova, G. S.. "Some classes of infinitely differentiable functions." Mathematica Bohemica 124.2-3 (1999): 167-172. <http://eudml.org/doc/248459>.

@article{Balashova1999,
abstract = {For nonquasianalytical Carleman classes conditions on the sequences $\lbrace \widehat\{M\}_n\rbrace $ and $\lbrace M_n\rbrace $ are investigated which guarantee the existence of a function in $C_J\lbrace \widehat\{M\}_n\rbrace $ such that u(n)(a) = bn,    bnKn+1Mn,    n = 0,1,...,    aJ. Conditions of coincidence of the sequences $\lbrace \widehat\{M\}_n\rbrace $ and $\lbrace M_n\rbrace $ are analysed. Some still unknown classes of such sequences are pointed out and a construction of the required function is suggested. The connection of this classical problem with the problem of the existence of a function with given trace at the boundary of the domain in a Sobolev space of infinite order is shown.},
author = {Balashova, G. S.},
journal = {Mathematica Bohemica},
keywords = {Carleman class; Sobolev space; Carleman class; Sobolev space},
language = {eng},
number = {2-3},
pages = {167-172},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Some classes of infinitely differentiable functions},
url = {http://eudml.org/doc/248459},
volume = {124},
year = {1999},
}

TY - JOUR
AU - Balashova, G. S.
TI - Some classes of infinitely differentiable functions
JO - Mathematica Bohemica
PY - 1999
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 124
IS - 2-3
SP - 167
EP - 172
AB - For nonquasianalytical Carleman classes conditions on the sequences $\lbrace \widehat{M}_n\rbrace $ and $\lbrace M_n\rbrace $ are investigated which guarantee the existence of a function in $C_J\lbrace \widehat{M}_n\rbrace $ such that u(n)(a) = bn,    bnKn+1Mn,    n = 0,1,...,    aJ. Conditions of coincidence of the sequences $\lbrace \widehat{M}_n\rbrace $ and $\lbrace M_n\rbrace $ are analysed. Some still unknown classes of such sequences are pointed out and a construction of the required function is suggested. The connection of this classical problem with the problem of the existence of a function with given trace at the boundary of the domain in a Sobolev space of infinite order is shown.
LA - eng
KW - Carleman class; Sobolev space; Carleman class; Sobolev space
UR - http://eudml.org/doc/248459
ER -

References

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  1. Mandelbroit S., Adjoining Series. Regularization of Sequences. Applications, Izdat. Inostrannoj Literatury, Moskva, 1995. (In Russian.) (1995) 
  2. Bang T, On quasi-analytiske funktioner, Thèse, København, 1946. (1946) 
  3. Borel E., Sur les fonctions d'une variable réelle indéfiniment dérivables, C.R. Acad. Sci. 174 (1922). (1922) 
  4. Carleman T., Les fonctions quasi-analytiques, Paris, 1926. (1926) 
  5. Carleson L., 10.7146/math.scand.a-10635, Math. Scand. 9 (1961), no. 2, 197-206. (1961) Zbl0114.05903MR0142012DOI10.7146/math.scand.a-10635
  6. Wahde G., 10.7146/math.scand.a-10815, Math. Scand. 20 (1967), no. 1, 19-31. (1967) MR0214976DOI10.7146/math.scand.a-10815
  7. Mitiagin B.S., On infinitely differentiable function with the values of its derivates given at a point, Dokl. Akad. Nauk SSSR 138 (1961), 289-292. (1961) MR0130946
  8. Ehrenpreis L., The punctual and local images of quasi-analytic and non-quasi-analytic classes, Institute for Advanced Study, Princeton, N. J., 1961. Mimeographed. (1961) 
  9. Balashova G. S., On extension of infinitely differentiable functions, Izv. Akad. Nauk SSSR, Ser. Mat. 51 (1987), no. 6, 1292-1308. (In Russian.) (1987) Zbl0643.26015MR0933965
  10. Balashova G. S., Conditions for the extension of a trace and an embedding for Banach spaces of infinitely differentiable functions, Mat. Sb. 184 (1993), no. 1, 105-128. (In Russian.) (1993) MR1211368
  11. Dubinskij Yu. A., Traces of functions from Sobolev spaces of infinite order and inhomogeneous problems for nonlinear equations, Mat. Sb. 106 (148) (1978), no. 1, 66-84. (In Russian.) (1978) Zbl0414.35028

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