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A unions and intersections of sets of synthesis. (English) Zbl 0839.43007

In this paper the Fourier algebra \(A (G)\) for a locally compact abelian group \(G\), and closed subsets \(E\) of \(G\) of synthesis or non-synthesis, are studied. The notations for equality, belonging, inclusion, etc. locally at a point \(x\), and the notion of spectrum of non-synthesis \(\Delta (E)\) of \(E\), introduced in the reviewer’s paper [Harmonic analysis, Lect. Notes Math. 781, 194-203 (1980; Zbl 0429.43008)], are frequently used. In that paper it is shown that if a closed subset \(E\) of \(G\) contains a \(C\)-set containing the non-synthesis spectrum \(\Delta (E)\), then the latter set is empty, i.e. \(E\) is of synthesis. In the present paper some rather straightforward consequences of this fact are obtained, the most interesting being some on infinite unions of synthesis sets.

MSC:

43A45 Spectral synthesis on groups, semigroups, etc.
43A46 Special sets (thin sets, Kronecker sets, Helson sets, Ditkin sets, Sidon sets, etc.)

Citations:

Zbl 0429.43008
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References:

[1] Colin C. Graham and O. Carruth McGehee, Essays in commutative harmonic analysis, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 238, Springer-Verlag, New York-Berlin, 1979. · Zbl 0439.43001
[2] Sadahiro Saeki, Spectral synthesis for the Kronecker sets, J. Math. Soc. Japan 21 (1969), 549 – 563. · Zbl 0211.15503 · doi:10.2969/jmsj/02140549
[3] D. L. Salinger and J. D. Stegeman, Difference spectra of ideals for nonmetrizable groups, J. London Math. Soc. (2) 26 (1982), no. 3, 531 – 540. · Zbl 0488.43010 · doi:10.1112/jlms/s2-26.3.531
[4] J. D. Stegeman, Some problems on spectral synthesis, Harmonic analysis, Iraklion 1978 (Proc. Conf., Univ. Crete, Iraklion, 1978), Lecture Notes in Math., vol. 781, Springer, Berlin, 1980, pp. 194 – 203.
[5] C. T. C. Wall, A geometric introduction to topology, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1972. · Zbl 0261.55001
[6] C. Robert Warner, A class of spectral sets, Proc. Amer. Math. Soc. 57 (1976), no. 1, 99 – 102. · Zbl 0309.43017
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