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Hilbert transform of $\mathrm{log}\vert f\vert$


Author: Javad Mashreghi
Journal: Proc. Amer. Math. Soc. 130 (2002), 683-688
MSC (2000): Primary 30D20; Secondary 42A50
Published electronically: July 31, 2001
MathSciNet review: 1866020
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Abstract:

There are two general ways to evaluate the Hilbert transform of a function of real variable $u(x)$. We can extend $u$ to a harmonic function in the upper half plane by the Poisson integral formula. Non-tangential limit of its harmonic conjugate exists almost everywhere and is defined to be the Hilbert transform of $u(x)$. There is also a singular integral formula for the Hilbert transform of $u(x)$. It is fairly difficult to directly evaluate the Hilbert transform of $u(x)$. In this paper we give an explicit formula for the Hilbert transform of $\log\vert f\vert$, where $f$ is a function in the Cartwright class.


References [Enhancements On Off] (What's this?)

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Additional Information

Javad Mashreghi
Affiliation: Department of Mathematics and Statistics, McGill University, Montreal, Canada H3A 2K6
Email: mashregh@math.mcgill.ca

DOI: https://doi.org/10.1090/S0002-9939-01-06335-3
Keywords: Entire functions of exponential type, Blaschke products, Hilbert transform
Received by editor(s): August 2, 2000
Published electronically: July 31, 2001
Additional Notes: This work was supported by Institut des sciences mathématiques (Montreal) and a J. W. McConnell McGill Major Fellowship.
Communicated by: Juha Heinonen
Article copyright: © Copyright 2001 American Mathematical Society