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## Studia Mathematica

2000 | 139 | 2 | 175-188
Tytuł artykułu

### On absolutely representing systems in spaces of infinitely differentiable functions

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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 $\overline K̇= K$. Moreover, absolutely representing systems of exponents in the space H(G) of functions analytic in an arbitrary domain $G ⊆ ℂ^p$ are also investigated.
Słowa kluczowe
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
175-188
Opis fizyczny
Daty
wydano
2000
otrzymano
1999-03-22
poprawiono
1999-12-28
Twórcy
autor
• Department of Mechanics and Mathematics, Rostov State University, ul. Zorge, 5, 344090 Rostov-na-Donu, Russia
Bibliografia
• [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).
• [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] A. V. Abanin, Weakly sufficient sets and absolutely representing systems, Doct. dissertation, Rostov-na-Donu, 1995, 268 pp. (in Russian).
• [4] Chan-Porn, Les systèmes de représentation absolue dans les espaces des fonctions holomorphes, Studia Math. 94 (1989), 193-212.
• [5] R. E. Edwards, Functional Analysis. Theory and Applications, Holt, Rinehart and Winston, New York, 1965.
• [6] E. Hille, Note on Dirichlet's series with complex exponents, Ann. of Math. 26 (1924), 261-278.
• [7] L. Hörmander, An Introduction to Complex Analysis in Several Variables, van Nostrand, Princeton, NJ, 1966.
• [8] L. Hörmander, The Analysis of Linear Partial Differential Operators. I. Distribution Theory and Fourier Analysis, Springer, 1983.
• [9] V. M. Kadets and Yu. F. Korobeĭnik, Representing and absolutely representing systems, Studia Math. 102 (1992), 217-223.
• [10] L. V. Kantorovich and G. P. Akilov, Functional Analysis in Normed Spaces, Fizmatgiz, Moscow, 1959 (in Russian); English transl., Macmillan, 1964.
• [11] Yu. F. Korobeĭnik, Representing systems, Math. USSR-Izv. 12 (1978), 309-335.
• [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] Yu. F. Korobeĭnik, Representing systems, Russian Math. Surveys 36 (1981), 75-137.
• [14] Yu. F. Korobeĭnik, Inductive and projective topologies. Sufficient sets and representing systems, Math. USSR-Izv. 28 (1987), 529-554.
• [15] Yu. F. Korobeĭnik, Absolutely representing families, Mat. Zametki 42 (1987), 670-680 (in Russian); English transl. in Soviet Math. Notes.
• [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] 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.
• 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.
• [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] 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] A. F. Leont'ev, Series of Exponentials, Nauka, Moscow, 1976 (in Russian).
• [22] A. Martineau, Sur la topologie des espaces de fonctions holomorphes, Math. Ann. 162 (1966), 68-88.
• [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.
• [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] 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] J. Sebastião e Silva, Su certe di spazi localmente convessi importanti per le applicazioni, Rend. Mat. Appl. 14 (1955), 388-410.
• [27] H. Whitney, Functions differentiable on the boundaries of regions, Ann. of Math. 33 (1934), 482-485.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-smv139i2p175bwm
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