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A space of slowly decreasing functions with pleasant Fourier transforms

Published online by Cambridge University Press:  14 November 2011

L. E. Fraenkel
L. E. Fraenkel
Affiliation:
School of Mathematics, University of Bath, Claverton Down, Bath BA2 7AY, U.K
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Synopsis

The space in question is Aµ(R):=L1(R) + Bµ(R), where Bµ(R) is a Banach space that contains the “tails” (the dominant parts for large values of |x|) of certain slowly decreasing functions from R to R. Functions in Bµ(R) are of bounded variation, and the norm involves their variation and a weighting function. Theorems are proved only for Bµ(R), because those for L1(R) are known. The results concern the convolution of a function in Bµ(R) with one in L1(R), the Fourier transform acting on Bµ(R), and the signum rule for the Hilbert transform of functions in Bµ(R).

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 119 , Issue 1-2 , 1991 , pp. 73 - 86
Copyright
Copyright © Royal Society of Edinburgh 1991

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