A compact set without Markov’s property but with an extension operator for C ∞ -functions

Alexander Goncharov

Studia Mathematica (1996)

  • Volume: 119, Issue: 1, page 27-35
  • ISSN: 0039-3223

Abstract

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We give an example of a compact set K ⊂ [0, 1] such that the space ℇ(K) of Whitney functions is isomorphic to the space s of rapidly decreasing sequences, and hence there exists a linear continuous extension operator L : ℇ ( K ) → C ∞ [ 0 , 1 ] . At the same time, Markov’s inequality is not satisfied for certain polynomials on K.

How to cite

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Goncharov, Alexander. "A compact set without Markov’s property but with an extension operator for $C^∞$-functions." Studia Mathematica 119.1 (1996): 27-35. <http://eudml.org/doc/216284>.

@article{Goncharov1996,
abstract = {We give an example of a compact set K ⊂ [0, 1] such that the space ℇ(K) of Whitney functions is isomorphic to the space s of rapidly decreasing sequences, and hence there exists a linear continuous extension operator $L: ℇ(K) → C^\{∞\}[0,1]$. At the same time, Markov’s inequality is not satisfied for certain polynomials on K.},
author = {Goncharov, Alexander},
journal = {Studia Mathematica},
keywords = {compact set; Whitney functions; rapidly decreasing sequences; linear continuous extension operator; Markov's inequality},
language = {eng},
number = {1},
pages = {27-35},
title = {A compact set without Markov’s property but with an extension operator for $C^∞$-functions},
url = {http://eudml.org/doc/216284},
volume = {119},
year = {1996},
}

TY - JOUR
AU - Goncharov, Alexander
TI - A compact set without Markov’s property but with an extension operator for $C^∞$-functions
JO - Studia Mathematica
PY - 1996
VL - 119
IS - 1
SP - 27
EP - 35
AB - We give an example of a compact set K ⊂ [0, 1] such that the space ℇ(K) of Whitney functions is isomorphic to the space s of rapidly decreasing sequences, and hence there exists a linear continuous extension operator $L: ℇ(K) → C^{∞}[0,1]$. At the same time, Markov’s inequality is not satisfied for certain polynomials on K.
LA - eng
KW - compact set; Whitney functions; rapidly decreasing sequences; linear continuous extension operator; Markov's inequality
UR - http://eudml.org/doc/216284
ER -

References

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  2. [2] L. P. Bos and P. D. Milman, On Markov and Sobolev type inequalities on compact sets in ℝ n , in: Topics in Polynomials of One and Several Variables and Their Applications, Th. M. Rassias, H. M. Srivastava and A. Yanushauskas (eds.), World Sci., 1993, 81-100. Zbl0865.46018
  3. [3] W. Pawłucki and W. Pleśniak, Extension of C ∞ functions from sets with polynomial cusps, Studia Math. 88 (1988), 279-287. Zbl0778.26010
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  6. [6] M. Tidten, Fortsetzungen von C ∞ -Funktionen, welche auf einer abgeschlossenen Menge in ℝ n definiert sind, Manuscripta Math. 27 (1979), 291-312. 
  7. [7] M. Tidten, Kriterien für die Existenz von Ausdehnungsoperatoren zu ℇ(K) für kompakte Teilmengen K von ℝ, Arch. Math. (Basel) 40 (1983), 73-81. 
  8. [8] A. F. Timan, Theory of Approximation of Functions of a Real Variable, Pergamon, Oxford, 1963. Zbl0117.29001
  9. [9] D. Vogt, Charakterisierung der Unterräume von s, Math. Z. 155 (1977), 109-117. Zbl0337.46015
  10. [10] D. Vogt, Sequence space representations of spaces of test functions and distributions, in: Functional Analysis, Holomorphy and Approximation Theory, G. I. Zapata (ed.), Lecture Notes in Pure and Appl. Math. 83, Dekker, 1983, 405-443. 
  11. [11] V. P. Zahariuta, Some linear topological invariants and isomorphisms of tensor products of scale's centers, Izv. Severo-Kavkaz. Nauchn. Tsentra Vyssh. Shkoly 4 (1974), 62-64 (in Russian). 
  12. [12] M. Zerner, Développement en séries de polynômes orthonormaux des fonctions indéfiniment différentiables, C. R. Acad. Sci. Paris 268 (1969), 218-220. Zbl0189.14601

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