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

 

A characterization of compact multipliers


Authors: Gregory F. Bachelis and Louis Pigno
Journal: Trans. Amer. Math. Soc. 165 (1972), 319-322
MSC: Primary 43A22; Secondary 43A25
MathSciNet review: 0300012
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Abstract: Let G be a compact abelian group and $ \varphi $ a complex-valued function defined on the dual $ \Gamma $. The main result of this paper is that $ \varphi $ is a compact multiplier of type $ (p,q),1 \leqq p < \infty $ and $ 1 \leqq q \leqq \infty $, if and only if it satisfies the following condition: Given $ \varepsilon > 0$ there corresponds a finite set $ K \subset \Gamma $ such that $ \vert\sum {a_\gamma }{b_\gamma }\varphi (\gamma )\vert < \varepsilon $ whenever $ P = \sum {a_\gamma }\gamma $ and $ Q = \sum {b_\gamma }\gamma $ are trigonometric polynomials satisfying $ {\left\Vert P \right\Vert _p} \leqq 1,{\left\Vert Q \right\Vert _{q'}} \leqq 1$ ($ q'$ the conjugate index of q) and $ {b_\gamma } = 0$ for $ \gamma \in K$. Using the above characterization we obtain the following necessary and sufficient condition for $ \varphi $ to be the Fourier transform of a continuous complex-valued function on G: Given $ \varepsilon > 0$ there corresponds a finite set $ K \subset \Gamma $ such that $ \vert\sum {b_\gamma }\varphi (\gamma )\vert < \varepsilon $ whenever $ Q = \sum {b_\gamma }\gamma $ is a trigonometric polynomial satisfying $ {\left\Vert Q \right\Vert _1} \leqq 1$ and $ {b_\gamma } = 0$ for $ \gamma \in K$.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1972-0300012-X
PII: S 0002-9947(1972)0300012-X
Article copyright: © Copyright 1972 American Mathematical Society