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Supports and singular supports of pseudomeasures

Published online by Cambridge University Press:  09 April 2009

R. E. Edwards
R. E. Edwards
Affiliation:
Department of Mathematics Institute of Advanced Studies Australian National University
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Summary

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Let G denote a Hausdorff locally compact Abelian group which is nondiscrete and second countable. The main results (Theorems (2.2) and (2.3)) assert that, for any closed subset E of G there exists a pseudomeasure s on G whose singular support is E; and that if no portion of E is a Helson set, then such an s may be chosen having its support equal to E. There follow (Corollaries (2.2.4) and (2.3.2)) sufficient conditions for the relations to hold for some pseudomeasure s, E and F being given closed subsets of G. These results are analogues and refinements of a theorem of Pollard [4] for the case G = R, which asserts the existence of a function in L∞(R) whose spectrum coincides with any preassigned closed subset of R.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 6 , Issue 1 , February 1966 , pp. 65 - 75
Copyright
Copyright © Australian Mathematical Society 1966

References

[1]Kahane, J.-P., et Salem, R.Ensembles parfaits et séries trigonométriques. Actualités Sci. et Ind. 1301. Paris (1963).Google Scholar
[2]Rudin, W.Fourier analysis on groups. Interscience Publishers (1962).Google Scholar
[3]Edwards, R. E.Functional Analysis: Theory and Applications. Holt, Rinehart and Winston Inc., New York (1965).Google Scholar
[4]Pollard, H.The harmonic analysis of bounded functions. Duke Math. J. 20 (1953), 499–512.CrossRefGoogle Scholar
[5]Gaudry, G. I. Multipliers of type (p, q). To appear, Pacific J. Math.Google Scholar
[6]Edwards, R. E.Spans of translates in L'(G). J. Australian Math. Soc. 5 (1965), 216–233.CrossRefGoogle Scholar
[7]Edwards, R. E. Uniform approximation on noncompact spaces. To appear, Trans. Amer. Math. Soc.Google Scholar