Idempotent multipliers on spaces of continuous functions with 𝑝-summable Fourier transforms

Bloom, Lynette M.; Bloom, Walter R.

doi:10.1090/S0002-9939-1980-0567988-7

Idempotent multipliers on spaces of continuous functions with $p$-summable Fourier transforms
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by Lynette M. Bloom and Walter R. Bloom PDF

Proc. Amer. Math. Soc. 79 (1980), 441-444 Request permission

Abstract:

Let G denote a compact abelian group, and ${A^p}$ the space of functions continuous on G and having p-summable Fourier transforms. The idempotent multipliers from ${A^p}$ to ${A^q}$ are characterised for $p,q \in [1,2]$.

References

Lynette M. Bloom, The Fourier multiplier problem for spaces of continuous functions with $p$-summable transforms, J. Austral. Math. Soc. 17 (1974), 319–331. Collection of articles dedicated to the memory of Hanna Neumann, VII. MR 0380236, DOI 10.1017/S1446788700017080
Lynette M. Butler, Certain non-algebras in harmonic analysis, Bull. Austral. Math. Soc. 4 (1971), 247–254. MR 284767, DOI 10.1017/S0004972700046529
Paul J. Cohen, On a conjecture of Littlewood and idempotent measures, Amer. J. Math. 82 (1960), 191–212. MR 133397, DOI 10.2307/2372731
Colin C. Graham, The Sidon constant of a finite abelian group, Proc. Amer. Math. Soc. 68 (1978), no. 1, 83–84. MR 458059, DOI 10.1090/S0002-9939-1978-0458059-X
Michael B. Gregory, $p$-Helson sets, $1<p<2$, Israel J. Math. 12 (1972), 356–368. MR 318792, DOI 10.1007/BF02764627
Edwin Hewitt and Gunter Ritter, Über die Integrierbarkeit von Fourier-Transformierten auf Gruppen. I. Stetige Funktionen mit kompaktem Träger und eine Bemerkung über hyperbolische Differentialoperatoren, Math. Ann. 224 (1976), no. 1, 77–96 (German). MR 420147, DOI 10.1007/BF01420289
Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. I, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 115, Springer-Verlag, Berlin-New York, 1979. Structure of topological groups, integration theory, group representations. MR 551496, DOI 10.1007/978-1-4419-8638-2