Abstract
Motivated by and extending a theorem of Reiter on sets of synthesis in \(\mathbb {R}^N\), we establish a general result for Fourier algebras of locally compact groups, even in the wider context of regular, semisimple and Tauberian commutative Banach algebras, which contains Reiter’s theorem as a special case and explains why it holds. In addition, we give a number of examples and several further results on weak spectral sets.
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References
Benedetto, J.J.: Spectral Synthesis. Teubner, Stuuttgart (1975)
Bröcker, T.H., tom Dieck, T.: Representations of Compact Lie Groups, Graduate Texts in Mathematics, vol. 98. Springer, New York (1985)
Degenfeld-Schonburg, S., Kaniuth, E., Lasser, R.: Spectral synthesis in Fourier algebras of ultraspherical hypergroups. J. Fourier Anal. Appl. 20, 258–281 (2014)
Eymard, P.: L’algèbre de Fourier d’un groupe localement compact. Bull. Soc. Math. France 92, 181–236 (1964)
Feinstein, J.F.: A non-trivial, strongly regular uniform algebra. J. Lond. Math. Soc. 45, 288–300 (1992)
Forrest, B.E.: Fourier analysis on coset spaces. Rocky Mt. J. Math. 28, 173–190 (1998)
Helgason, S.: Differential Geometry, Lie Groups and Symmetric Spaces. Academic Press, New York (1978)
Herz, C.: Spectral synthesis for circles. Ann. Math. 68, 709–712 (1958)
Kaniuth, E.: Weak spectral synthesis in commutative Banach algebras. J. Funct. Anal. 254, 987–1002 (2008)
Kaniuth, E.: A Course in Commutative Banach Algebras, Graduate Texts in Mathematics, vol. 246. Springer, New York (2009)
Kaniuth, E.: Weak spectral synthesis in commutative Banach algebras. II. J. Funct. Anal. 259, 524–544 (2010)
Kaniuth, E.: Weak spectral synthesis in Fourier algebras of coset spaces. Studia Math. 197, 229–246 (2010)
Kaniuth, E., Ülger, A.: Weak spectral synthesis in commutative Banach algebras. III. J. Funct. Anal. 268, 2142–2170 (2015)
Lust, F.: Le problème de la synthèse et de la \(p\)-finesse pour certaines orbites de groupes linéaires dans \(A_p(\mathbb{R}^n)\). Studia Math. 39, 17–28 (1971)
Meaney, C.: On the failure of spectral synthesis for compact semisimple Lie groups. J. Funct. Anal. 48, 43–57 (1982)
Muraleedharan, M., Parthasarathy, K.: Difference spectrum and spectral synthesis. Tohoku Math. J. 51, 65–73 (1999)
Parthasarathy, K., Prakash, R.: Malliavin’s theorem for weak synthesis on nonabelian groups. Bull. Sci. math. 134, 561–578 (2010)
Parthasarathy, K., Shravan Kumar, N.: Fourier algebras on homogeneous spaces. Bull. Sci. Math. 135, 187–205 (2011)
Parthasarathy, K., Varma, S.: On weak spectral synthesis. Bull. Aust. Math. Soc. 43, 279–282 (1991)
Reiter, H.: Contributions to harmonic analysis. IV. Math. Ann. 135, 467–476 (1958)
Reiter, H., Stegeman, J.D.: Classical Harmonic Analysis and Locally Compact Groups. Oxford University Press, Oxford (2000)
Rudin, W.: Fourier analysis on groups. Interscience, New York (1962)
Saeki, S.: On Helson sets that disobey spectral synthesis. Proc. Am. Math. Soc. 47, 71–77 (1975)
Ülger, A.: Relatively weak\(^\ast \)-closed ideals of \(A(G)\), sets of synthesis and sets of uniqueness. Colloq. Math. 136, 271–296 (2014)
Varopoulos, NTh: Spectral synthesis on spheres. Proc. Camb. Phil. Soc. 62, 379–387 (1966)
Warner, C.R.: Weak spectral synthesis. Proc. Am. Math. Soc. 99, 244–248 (1987)
Warner, C.R.: Weak \(s\)-sets and Helson sets. Contemp. Math. 91, 291–294 (1989)
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Communicated by A. Constantin.
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Kaniuth, E., Ülger, A. On a theorem of Reiter and spectral synthesis. Monatsh Math 181, 839–853 (2016). https://doi.org/10.1007/s00605-016-0885-1
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DOI: https://doi.org/10.1007/s00605-016-0885-1
Keywords
- Locally compact group
- Fourier algebra
- Set of synthesis
- Ditkin set
- Commutative Banach algebra
- Structure space
- Weak spectral set