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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Functions which are restrictions of $ L\sp{p}$-multipliers


Author: Michael G. Cowling
Journal: Trans. Amer. Math. Soc. 213 (1975), 35-51
MSC: Primary 43A22
MathSciNet review: 0390653
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Abstract: Raouf Doss has given a sufficient condition for a measurable function $ \phi $ on a measurable subset $ \Lambda $ of an LCA group $ \Gamma $ to be the restriction (l.a.e.) to $ \Lambda $ of the Fourier transform of a bounded measure, i.e., a Fourier multiplier of type (1, 1). We generalise Doss' theorem, and prove that, if the measurable function $ \phi $ on $ \Lambda $ is approximable on finite subsets of $ \Lambda $ by trigonometric polynomials which are Fourier multipliers of type (p, p) on $ \Gamma $ of norms no greater than C, then $ \phi $ is equal locally almost everywhere to the restriction to $ \Lambda $ of a Fourier multiplier of type (p, p) and norm no greater than C.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1975-0390653-9
PII: S 0002-9947(1975)0390653-9
Keywords: Locally compact abelian group, multipliers, restriction of a multiplier, approximation
Article copyright: © Copyright 1975 American Mathematical Society