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Transactions of the American Mathematical Society

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

 

 

Weak restricted and very restricted operators on $ L\sp{2}$


Author: J. Marshall Ash
Journal: Trans. Amer. Math. Soc. 281 (1984), 675-689
MSC: Primary 42A45; Secondary 42A50, 47B38
MathSciNet review: 722768
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Abstract: A battlement is a real function with values in $ \{ 0,1\} $ that looks like a castle battlement. A commuting with translation linear operator $ T$ mapping step functions on $ {\mathbf{R}}$ into the set of all measurable functions on $ {\mathbf{R}}$ and satisfying $ \parallel Tb{\parallel_2} \leqslant C\parallel b{\parallel_2}$ for all battlements $ b$ is bounded on $ {L^2}({\mathbf{R}})$. This remains true if the underlying space is the circle but is demonstrably false if the underlying space is the integers. Michael Cowling's theorem that linear commuting with translation operators are bounded on $ {L^2}$ if they are weak restricted $ (2,2)$ is reproved and an application of this result to sums of exponentials is given.


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DOI: https://doi.org/10.1090/S0002-9947-1984-0722768-2
Keywords: Operator, weak restricted type $ (2,2)$, commuting with translation, convolution operator, multiplier
Article copyright: © Copyright 1984 American Mathematical Society