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On Positivity of Fourier Transforms

Published online by Cambridge University Press:  17 April 2009

E.O. Tuck
E.O. Tuck
Affiliation:
Applied MathematicsThe University of AdelaideAustralia5005
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This note concerns Fourier transforms on the real positive line. In particular, we seek conditions on a real function u(x) in x > 0, that ensure that its Fourier-cosine transform v(t) = u(x) cos xt dx is positive. We prove first that this is so for all t > 0, if u"(x) > 0 for all x > 0, that is, that everywhere-convex functions have everywhere-positive Fourier-cosine transforms. We then obtain a complex-plane criterion for some types of non-convex u(x). Finally we consider criteria on u(x) that imply positivity of v(t) for t > t0, for some t0 > 0.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 74 , Issue 1 , August 2006 , pp. 133 - 138
Copyright
Copyright © Australian Mathematical Society 2006

References

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