Infinite systems of linear equations for real analytic functions
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- by P. Domański and D. Vogt PDF
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Abstract:
We study the problem when an infinite system of linear functional equations \[ \mu _n(f)=b_n\quad \text {for\ }n\in \mathbb {N}\] has a real analytic solution $f$ on $\omega \subseteq \mathbb {R}^d$ for every right-hand side $(b_n)_{n\in \mathbb {N}}\subseteq \mathbb {C}$ and give a complete characterization of such sequences of analytic functionals $(\mu _n)$. We also show that every open set $\omega \subseteq \mathbb {R}^d$ has a complex neighbourhood $\Omega \subseteq \mathbb {C}^d$ such that the positive answer is equivalent to the positive answer for the analogous question with solutions holomorphic on $\Omega$.References
- S. Banach, Théorie des Opérations Linéaires, Monografie Matematyczne, vol. 1, Warszawa, 1932.
- José Bonet, PawełDomański, and Dietmar Vogt, Interpolation of vector-valued real analytic functions, J. London Math. Soc. (2) 66 (2002), no. 2, 407–420. MR 1920411, DOI 10.1112/S0024610702003551
- Henri Cartan, Variétés analytiques réelles et variétés analytiques complexes, Bull. Soc. Math. France 85 (1957), 77–99 (French). MR 94830
- P. Domański, L. Frerick, D. Vogt, Fréchet quotients of spaces of real analytic functions, Studia Math. 159 (2003), 229-245.
- PawełDomański and Michael Langenbruch, Composition operators on spaces of real analytic functions, Math. Nachr. 254/255 (2003), 68–86. MR 1983956, DOI 10.1002/mana.200310053
- P. Domański and D. Vogt, A splitting theory for the space of distributions, Studia Math. 140 (2000), no. 1, 57–77. MR 1763882, DOI 10.4064/sm-140-1-57-77
- PawełDomański and Dietmar Vogt, The space of real-analytic functions has no basis, Studia Math. 142 (2000), no. 2, 187–200. MR 1792604, DOI 10.4064/sm-142-2-187-200
- —, A non-trivial Fréchet quotient of the space of real analytic functions, Arch. Math. 81 (2003), 208-214.
- M. Eidelheit, Zur Theorie der Systeme linearer Gleichungen, Studia Math. 6 (1936), 139-148.
- Hans Grauert, On Levi’s problem and the imbedding of real-analytic manifolds, Ann. of Math. (2) 68 (1958), 460–472. MR 98847, DOI 10.2307/1970257
- Francesco Guaraldo, Patrizia Macrì, and Alessandro Tancredi, Topics on real analytic spaces, Advanced Lectures in Mathematics, Friedr. Vieweg & Sohn, Braunschweig, 1986. MR 1013362, DOI 10.1007/978-3-322-84243-5
- Gottfried Köthe, Topological vector spaces. I, Die Grundlehren der mathematischen Wissenschaften, Band 159, Springer-Verlag New York, Inc., New York, 1969. Translated from the German by D. J. H. Garling. MR 0248498
- Steven G. Krantz, Function theory of several complex variables, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1982. MR 635928
- Michael Langenbruch, Analytic extension of smooth functions, Results Math. 36 (1999), no. 3-4, 281–296. MR 1726219, DOI 10.1007/BF03322117
- André Martineau, Sur les fonctionnelles analytiques et la transformation de Fourier-Borel, J. Analyse Math. 11 (1963), 1–164 (French). MR 159220, DOI 10.1007/BF02789982
- A. Martineau, Sur la topologie des espaces de fonctions holomorphes, Math. Ann. 163 (1966), 62–88 (French). MR 190697, DOI 10.1007/BF02052485
- Reinhold Meise and Dietmar Vogt, Introduction to functional analysis, Oxford Graduate Texts in Mathematics, vol. 2, The Clarendon Press, Oxford University Press, New York, 1997. Translated from the German by M. S. Ramanujan and revised by the authors. MR 1483073
- B. S. Mitjagin, Approximate dimension and bases in nuclear spaces, Uspehi Mat. Nauk 16 (1961), no. 4 (100), 63–132 (Russian). MR 0152865
- Pierre Schapira, Théorie des hyperfonctions, Lecture Notes in Mathematics, Vol. 126, Springer-Verlag, Berlin-New York, 1970 (French). MR 0631543
- Dietmar Vogt, Kernels of Eidelheit matrices and related topics, Proceedings of the functional analysis conference (Silivri/Istanbul, 1985), 1986, pp. 232–256 (English, with Turkish summary). MR 872881
- Dietmar Vogt, On two problems of Mityagin, Math. Nachr. 141 (1989), 13–25. MR 1014410, DOI 10.1002/mana.19891410103
- D. Vogt, Kernels of power series matrices, Trudy Mat. Inst. Steklov. 203 (1994), no. Izbran. Voprosy Mat. Fiz. i Anal., 461–468; English transl., Proc. Steklov Inst. Math. 3(203) (1995), 377–381. MR 1382621
- Kôsaku Yosida, Functional analysis, 4th ed., Die Grundlehren der mathematischen Wissenschaften, Band 123, Springer-Verlag, New York-Heidelberg, 1974. MR 0350358
Additional Information
- P. Domański
- Affiliation: Faculty of Mathematics and Computer Science, A. Mickiewicz University Poznań and Institute of Mathematics, Polish Academy of Sciences (Poznań branch), ul. Umultowska 87, 61-614 Poznań, Poland
- Email: domanski@amu.edu.pl
- D. Vogt
- Affiliation: Bergische Universität Wuppertal, FB Mathematik, Gaußstr. 20, D–42097 Wuppertal, Germany
- MR Author ID: 179065
- Email: dvogt@math.uni-wuppertal.de
- Received by editor(s): January 28, 2003
- Received by editor(s) in revised form: May 22, 2003, and July 9, 2003
- Published electronically: July 20, 2004
- Communicated by: N. Tomczak-Jaegermann
- © Copyright 2004 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 132 (2004), 3607-3614
- MSC (2000): Primary 46E10; Secondary 46A13, 26E05, 46F15
- DOI: https://doi.org/10.1090/S0002-9939-04-07435-0
- MathSciNet review: 2084083