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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Cesàro means of Fourier transforms and multipliers on $ L\sp 1({\bf R})$

Authors: Dăng Vũ Giang and Ferenc Móricz
Journal: Proc. Amer. Math. Soc. 122 (1994), 469-477
MSC: Primary 42A45; Secondary 42A38
MathSciNet review: 1201804
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Abstract: We prove that the Cesàro mean $ \sigma $ of a multiplier $ \lambda $ on $ {L^1}({\mathbf{R}})$ is also a multiplier on $ {L^1}({\mathbf{R}})$. In the particular cases when (i) $ \lambda $ is odd, we prove that $ \sigma $ is the Fourier transform of an odd function in the Hardy space $ {H^1}({\mathbf{R}})$, and (ii) $ \lambda $ is even, we give a necessary and sufficient condition in order that $ \sigma $ be a Fourier transform of an even function in $ {L^1}({\mathbf{R}})$. As a corollary, we obtain a nontrivial condition for $ \lambda $ in order to be a multiplier on $ {L^1}({\mathbf{R}})$; namely,

$\displaystyle \int_0^\infty {\left\vert {\frac{1}{t}\int_0^t {\{ \lambda (\xi ) - \lambda ( - \xi )\} \,d\xi } } \right\vert} \frac{{dt}}{t} < \infty .$

We also prove Hardy type inequalities for multipliers and Hilbert transforms.

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Keywords: Fourier transform, Hilbert transform, Hardy space $ {H^1}({\mathbf{R}})$, multiplier, function of bounded variation, Fourier-Stieltjes series, conjugate series, Cesàro mean, arithmetic mean, Hardy type inequality
Article copyright: © Copyright 1994 American Mathematical Society