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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Approximations and representations for Fourier transforms

Author: Raouf Doss
Journal: Trans. Amer. Math. Soc. 153 (1971), 211-221
MSC: Primary 42.52
MathSciNet review: 0268597
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Abstract: $ G$ is a locally compact abelian group with dual $ \Gamma $. If $ p(\gamma ) = \sum\nolimits_1^N {{a_n}({x_n},\gamma )} $ is a trigonometric polynomial, its capacity, by definition is $ \Sigma \vert{a_n}\vert$. The main theorem is: Let $ \varphi $ be a measurable function defined on the measurable subset $ \Lambda $ of $ \Gamma $. If $ \varphi $ can be approximated on finite sets in $ \Lambda $ by trigonometric polynomials of capacity at most $ C$ (constant), then $ \varphi = \hat \mu $, locally almost everywhere on $ \Lambda $, where $ \mu $ is a regular bounded measure on $ G$ and $ \vert\vert\mu \vert\vert \leqq C$.

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Keywords: Locally compact abelian group, Fourier-Stieltjes transform of a measure, trigonometric polynomials, approximation
Article copyright: © Copyright 1971 American Mathematical Society