Spectral synthesis in the space of functions of exponential growth on a finitely generated Abelian group
HTML articles powered by AMS MathViewer
- by
S. S. Platonov
Translated by: S. Kislyakov - St. Petersburg Math. J. 24 (2013), 663-675
- DOI: https://doi.org/10.1090/S1061-0022-2013-01259-3
- Published electronically: May 24, 2013
- PDF | Request permission
Abstract:
It is shown that spectral synthesis occurs in the space of functions of exponential growth on an arbitrary finitely-generated discrete Abelian group.References
- Laurent Schwartz, Théorie générale des fonctions moyenne-périodiques, Ann. of Math. (2) 48 (1947), 857–929 (French). MR 23948, DOI 10.2307/1969386
- John E. Gilbert, On the ideal structure of some algebras of analytic functions, Pacific J. Math. 35 (1970), 625–634. MR 412439
- S. S. Platonov, Spectral synthesis in some function topological vector spaces, Algebra i Analiz 22 (2010), no. 5, 154–185 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 22 (2011), no. 5, 813–833. MR 2828831, DOI 10.1090/S1061-0022-2011-01170-7
- D. I. Gurevič, Counterexamples to a problem of L. Schwartz, Funkcional. Anal. i Priložen. 9 (1975), no. 2, 29–35 (Russian). MR 0390759
- Laurent Schwartz, Analyse et synthèse harmoniques dans les espaces de distributions, Canad. J. Math. 3 (1951), 503–512 (French). MR 44754, DOI 10.4153/cjm-1951-051-5
- László Székelyhidi, Discrete spectral synthesis and its applications, Springer Monographs in Mathematics, Springer, Dordrecht, 2006. MR 2279454
- Marcel Lefranc, Analyse spectrale sur $Z_{n}$, C. R. Acad. Sci. Paris 246 (1958), 1951–1953 (French). MR 98951
- László Székelyhidi, On discrete spectral synthesis, Functional equations—results and advances, Adv. Math. (Dordr.), vol. 3, Kluwer Acad. Publ., Dordrecht, 2002, pp. 263–274. MR 1912720
- Áron Bereczky and László Székelyhidi, Spectral synthesis on torsion groups, J. Math. Anal. Appl. 304 (2005), no. 2, 607–613. MR 2126554, DOI 10.1016/j.jmaa.2004.09.046
- Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976. MR 0482275
- A. P. Robertson and W. J. Robertson, Topological vector spaces, Cambridge Tracts in Mathematics and Mathematical Physics, No. 53, Cambridge University Press, New York, 1964. MR 0162118
- N. K. Nikol′skiĭ, Invariant subspaces in operator theory and function theory, Mathematical analysis, Vol. 12 (Russian), Akad. Nauk SSSR Vsesojuz. Inst. Naučn. i Tehn. Informacii, Moscow, 1974, pp. 199–412, 468. (loose errata) (Russian). MR 0430821
- A. G. Baskakov, Theory of representations of Banach algebras, and abelian groups and semigroups in the spectral analysis of linear operators, Sovrem. Mat. Fundam. Napravl. 9 (2004), 3–151 (Russian); English transl., J. Math. Sci. (N.Y.) 137 (2006), no. 4, 4885–5036. MR 2123307, DOI 10.1007/s10958-006-0286-4
- V. V. Napalkov, Uravneniya svertki v mnogomernykh prostranstvakh, “Nauka”, Moscow, 1982 (Russian). MR 678923
- Oswald Teichmüller, Braucht der Algebraiker das Auswahlaxiom?, Deutsche Math. 4 (1939), 567–577 (German). MR 212
- H. Cartan, Seminaire Henri Cartan. 1951/1952. Fonctions analytiques de plusieurs variables complexes, W. A. Benjamin, Inc., New York–Amsterdam, 1967.
- Henri Cartan, Idéaux et modules de fonctions analytiques de variables complexes, Bull. Soc. Math. France 78 (1950), 29–64 (French). MR 36848
- Robert C. Gunning and Hugo Rossi, Analytic functions of several complex variables, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965. MR 0180696
- B. V. Shabat, Vvedenie v kompleksnyĭ analiz. Chast′II. Funktsii neskol′kikh peremennykh, Izdat. “Nauka”, Moscow, 1976 (Russian). Second edition, revised and augmented. MR 0584935
Bibliographic Information
- S. S. Platonov
- Affiliation: Petrozavodsk State University, Prospect Lenina 33, Petrozavodsk 185910, Russia
- Email: platonov@psu.karelia.ru
- Received by editor(s): September 9, 2011
- Published electronically: May 24, 2013
- © Copyright 2013 American Mathematical Society
- Journal: St. Petersburg Math. J. 24 (2013), 663-675
- MSC (2010): Primary 43A45
- DOI: https://doi.org/10.1090/S1061-0022-2013-01259-3
- MathSciNet review: 3088012