On absolutely representing systems in spaces of infinitely differentiable functions

Yu. Korobeĭnik

Studia Mathematica (2000)

  • Volume: 139, Issue: 2, page 175-188
  • ISSN: 0039-3223

Abstract

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The main part of the paper is devoted to the problem of the existence of absolutely representing systems of exponentials with imaginary exponents in the spaces C ∞ ( G ) and C ∞ ( K ) of infinitely differentiable functions where G is an arbitrary domain in ℝ p , p≥1, while K is a compact set in ℝ p with non-void interior K̇ such that K ¯ ̇ = K . Moreover, absolutely representing systems of exponents in the space H(G) of functions analytic in an arbitrary domain G ⊆ ℂ p are also investigated.

How to cite

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Korobeĭnik, Yu.. "On absolutely representing systems in spaces of infinitely differentiable functions." Studia Mathematica 139.2 (2000): 175-188. <http://eudml.org/doc/216717>.

@article{Korobeĭnik2000,
abstract = {The main part of the paper is devoted to the problem of the existence of absolutely representing systems of exponentials with imaginary exponents in the spaces $C^∞(G)$ and $C^∞(K)$ of infinitely differentiable functions where G is an arbitrary domain in $ℝ^p$, p≥1, while K is a compact set in $ℝ^p$ with non-void interior K̇ such that $\overline\{K\}̇= K$. Moreover, absolutely representing systems of exponents in the space H(G) of functions analytic in an arbitrary domain $G ⊆ ℂ^p$ are also investigated.},
author = {Korobeĭnik, Yu.},
journal = {Studia Mathematica},
keywords = {infinitely differentiable functions; absolutely representing system; Whitney compactum},
language = {eng},
number = {2},
pages = {175-188},
title = {On absolutely representing systems in spaces of infinitely differentiable functions},
url = {http://eudml.org/doc/216717},
volume = {139},
year = {2000},
}

TY - JOUR
AU - Korobeĭnik, Yu.
TI - On absolutely representing systems in spaces of infinitely differentiable functions
JO - Studia Mathematica
PY - 2000
VL - 139
IS - 2
SP - 175
EP - 188
AB - The main part of the paper is devoted to the problem of the existence of absolutely representing systems of exponentials with imaginary exponents in the spaces $C^∞(G)$ and $C^∞(K)$ of infinitely differentiable functions where G is an arbitrary domain in $ℝ^p$, p≥1, while K is a compact set in $ℝ^p$ with non-void interior K̇ such that $\overline{K}̇= K$. Moreover, absolutely representing systems of exponents in the space H(G) of functions analytic in an arbitrary domain $G ⊆ ℂ^p$ are also investigated.
LA - eng
KW - infinitely differentiable functions; absolutely representing system; Whitney compactum
UR - http://eudml.org/doc/216717
ER -

References

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  1. [1] A. V. Abanin, On continuation and stability of weakly sufficient sets, Izv. Vyssh. Uchebn. Zaved. Mat. 4 (1987), 3-10 (in Russian); English transl. in Soviet Math. (Izv. VUZ). Zbl0623.32004
  2. [2] A. V. Abanin, Representation of functions by series of exponentials and universal classes of convex domains, in: Linear Operators in Complex Analysis, O. V. Epifanov (ed.), Rostov State Univ. Press, 1994, 3-9 (in Russian). 
  3. [3] A. V. Abanin, Weakly sufficient sets and absolutely representing systems, Doct. dissertation, Rostov-na-Donu, 1995, 268 pp. (in Russian). 
  4. [4] Chan-Porn, Les systèmes de représentation absolue dans les espaces des fonctions holomorphes, Studia Math. 94 (1989), 193-212. Zbl0704.46014
  5. [5] R. E. Edwards, Functional Analysis. Theory and Applications, Holt, Rinehart and Winston, New York, 1965. Zbl0182.16101
  6. [6] E. Hille, Note on Dirichlet's series with complex exponents, Ann. of Math. 26 (1924), 261-278. Zbl50.0233.03
  7. [7] L. Hörmander, An Introduction to Complex Analysis in Several Variables, van Nostrand, Princeton, NJ, 1966. Zbl0138.06203
  8. [8] L. Hörmander, The Analysis of Linear Partial Differential Operators. I. Distribution Theory and Fourier Analysis, Springer, 1983. Zbl0521.35001
  9. [9] V. M. Kadets and Yu. F. Korobeĭnik, Representing and absolutely representing systems, Studia Math. 102 (1992), 217-223. Zbl0811.46005
  10. [10] L. V. Kantorovich and G. P. Akilov, Functional Analysis in Normed Spaces, Fizmatgiz, Moscow, 1959 (in Russian); English transl., Macmillan, 1964. Zbl0127.06102
  11. [11] Yu. F. Korobeĭnik, Representing systems, Math. USSR-Izv. 12 (1978), 309-335. 
  12. [12] Yu. F. Korobeĭnik, On representing systems, in: Current Questions in Mathematical Analysis, K. K. Mokrishchev and V. P. Zaharjuta (ed.), Rostov State Univ. Press, 1978, 100-111 (in Russian). 
  13. [13] Yu. F. Korobeĭnik, Representing systems, Russian Math. Surveys 36 (1981), 75-137. Zbl0483.30003
  14. [14] Yu. F. Korobeĭnik, Inductive and projective topologies. Sufficient sets and representing systems, Math. USSR-Izv. 28 (1987), 529-554. Zbl0628.46004
  15. [15] Yu. F. Korobeĭnik, Absolutely representing families, Mat. Zametki 42 (1987), 670-680 (in Russian); English transl. in Soviet Math. Notes. Zbl0639.46009
  16. [16] Yu. F. Korobeĭnik, On the Cauchy problem for linear systems with variable coefficients, manuscript, Rostov-na-Donu, 1997, VINITI 2501-B97, 64 pp.; Referat. Zh. Mat. 1998, no. 1, ref. 1B337 (in Russian). 
  17. [17] Yu. F. Korobeĭnik, Representing systems of exponentials and the Cauchy problem for partial differential equations with constant coefficients, Izv. Ross. Akad. Nauk Ser. Mat. 61 (1997), no. 3, 91-132 (in Russian); English transl.: Izv. Math. 61 (1997), 553-592. 
  18. Yu. F. Korobeĭnik, Absolutely convergent Dirichlet series and analytic continuation of its sum, Lobachevski J. Math. 1 (1998), 15-44; http://www.kcn.ru/tat_en/science/ljm/ contents.html. Zbl0932.30003
  19. [19] Yu. F. Korobeĭnik and A. F. Leont'ev, On the property of inner continuation of representing systems of exponentials, Mat. Zametki 28 (1980), 243-254 (in Russian); English transl. in Soviet Math. Notes. 
  20. [20] Yu. F. Korobeĭnik and A. B. Mikhailov, Analytic solutions of the Cauchy problem, Differentsial'nye Uravneniya 27 (1991), 503-510 (in Russian); English. transl.: Differential Equations 27 (1991), 361-366. 
  21. [21] A. F. Leont'ev, Series of Exponentials, Nauka, Moscow, 1976 (in Russian). 
  22. [22] A. Martineau, Sur la topologie des espaces de fonctions holomorphes, Math. Ann. 162 (1966), 68-88. Zbl0138.38101
  23. [23] R. Meise and B. A. Taylor, Linear extension operators for ultradifferentiable functions of Beurling type on compact sets, Amer. J. Math. 111 (1989), 309-337. Zbl0696.46001
  24. [24] V. V. Morzhakov, Absolutely representing systems of exponentials in the space of analytic functions in several variables, manuscript, Rostov-na-Donu, 1981, VINITI 245-81, 30 pp.; Referat. Zh. Mat. 1981, no. 4, 4B109 (in Russian). 
  25. [25] V. V. Morzhakov, Absolutely representing systems in the spaces of analytic functions in several variables, in: Theory of Functions and Approximations, Works of Saratov Winter School, Saratov State Univ. Press, part 2, 1983, 92-94 (in Russian). 
  26. [26] J. Sebastião e Silva, Su certe di spazi localmente convessi importanti per le applicazioni, Rend. Mat. Appl. 14 (1955), 388-410. Zbl0064.35801
  27. [27] H. Whitney, Functions differentiable on the boundaries of regions, Ann. of Math. 33 (1934), 482-485. Zbl60.0217.03

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