Spectral synthesis in the space of functions of exponential growth on a finitely generated Abelian group

Platonov, S.

doi:10.1090/S1061-0022-2013-01259-3

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