A remark on the approximation theorems of Whitney and Carleman-Scheinberg

Michal Johanis

Commentationes Mathematicae Universitatis Carolinae (2015)

  • Volume: 56, Issue: 1, page 1-6
  • ISSN: 0010-2628

Abstract

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We show that a C k -smooth mapping on an open subset of ℝ n , k ∈ ℕ ∪ { 0 , ∞ } , can be approximated in a fine topology and together with its derivatives by a restriction of a holomorphic mapping with explicitly described domain. As a corollary we obtain a generalisation of the Carleman-Scheinberg theorem on approximation by entire functions.

How to cite

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Johanis, Michal. "A remark on the approximation theorems of Whitney and Carleman-Scheinberg." Commentationes Mathematicae Universitatis Carolinae 56.1 (2015): 1-6. <http://eudml.org/doc/269884>.

@article{Johanis2015,
abstract = {We show that a $C^k$-smooth mapping on an open subset of $\mathbb \{R\}^n$, $k\in \mathbb \{N\}\cup \lbrace 0,\infty \rbrace $, can be approximated in a fine topology and together with its derivatives by a restriction of a holomorphic mapping with explicitly described domain. As a corollary we obtain a generalisation of the Carleman-Scheinberg theorem on approximation by entire functions.},
author = {Johanis, Michal},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {approximation; real-analytic; entire functions; approximation; real-analytic; entire functions},
language = {eng},
number = {1},
pages = {1-6},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {A remark on the approximation theorems of Whitney and Carleman-Scheinberg},
url = {http://eudml.org/doc/269884},
volume = {56},
year = {2015},
}

TY - JOUR
AU - Johanis, Michal
TI - A remark on the approximation theorems of Whitney and Carleman-Scheinberg
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2015
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 56
IS - 1
SP - 1
EP - 6
AB - We show that a $C^k$-smooth mapping on an open subset of $\mathbb {R}^n$, $k\in \mathbb {N}\cup \lbrace 0,\infty \rbrace $, can be approximated in a fine topology and together with its derivatives by a restriction of a holomorphic mapping with explicitly described domain. As a corollary we obtain a generalisation of the Carleman-Scheinberg theorem on approximation by entire functions.
LA - eng
KW - approximation; real-analytic; entire functions; approximation; real-analytic; entire functions
UR - http://eudml.org/doc/269884
ER -

References

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  1. Carleman T., Sur un théorème de Weierstrass, (French), Ark. Mat., Ast. Fysik 20B (1927), no. 4, 1–5. 
  2. Frih E.M., Gauthier P.M., 10.4153/CMB-1988-071-1, Canad. Math. Bull. 31 (1988), no. 4, 495–499. Zbl0641.32011MR0971578DOI10.4153/CMB-1988-071-1
  3. Fabian M., Habala P., Hájek P., Montesinos V., Zizler V., Banach Space Theory, CMS Books in Mathematics, Springer, New York, 2011. Zbl1229.46001MR2766381
  4. Hille E., Functional analysis and semi-groups, Amer. Math. Soc. Colloq. Publ. 31, American Mathematical Society, New York, 1948. Zbl0392.46001MR0025077
  5. Hoischen L., 10.1016/0021-9045(73)90093-2, (German), J. Approx. Theory 9 (1973), no. 3, 272–277. Zbl0271.30035MR0367217DOI10.1016/0021-9045(73)90093-2
  6. Kaplan W., 10.1307/mmj/1031710533, Michigan Math. J. 3 (1955), no. 1, 43–52. Zbl0070.06203MR0070753DOI10.1307/mmj/1031710533
  7. Nersesyan A., On Carleman sets, Amer. Math. Soc. Transl. Ser. 2 122 (1984), 99–104. Zbl0552.30029
  8. Scheinberg S., 10.1007/BF02789974, J. Analyse Math. 29 (1976), 16–18. Zbl0343.41022MR0508100DOI10.1007/BF02789974
  9. Whitney H., 10.1090/S0002-9947-1934-1501735-3, Trans. Amer. Math. Soc. 36 (1934), no. 1, 63–89. Zbl0008.24902MR1501735DOI10.1090/S0002-9947-1934-1501735-3

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