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The positive part of a Fourier transform

Part of: Harmonic analysis in several variables

Published online by Cambridge University Press:  09 April 2009

Augusto Logli
Augusto Logli
Affiliation:
Istituto di Matematica Universita di GenovaVia L.B. Alberti 4 16132, Genova, Italy
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Abstract

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We consider the function u whose Fourier transform is the positive part of the Fourier transform of a function f on Rn. If n ≤ 2 and f satisfies simple regularity conditions (in particular if f is in the Schwartz space Y(Rn), then u lies in L1(Rn). If n ≥ 3, then simple counterexamples exist; for example, if f(x) = |x|2 exp(-|x|2), then u does not lie in L1(Rn).

MSC classification

Secondary: 42B10: Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 38 , Issue 2 , April 1985 , pp. 152 - 156
Copyright
Copyright © Australian Mathematical Society 1985

References

[1]Reiter, H., Classical harmonic analysis and locally compact groups (Oxford University Press, Oxford, 1968).Google Scholar