Hilbert transform of $\mathrm {log}|f|$
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- by Javad Mashreghi PDF
- Proc. Amer. Math. Soc. 130 (2002), 683-688 Request permission
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 $\log |f|$, where $f$ is a function in the Cartwright class.References
- Paul Koosis, The logarithmic integral. I, Cambridge Studies in Advanced Mathematics, vol. 12, Cambridge University Press, Cambridge, 1988. MR 961844, DOI 10.1017/CBO9780511566196
- John B. Conway, Functions of one complex variable, 2nd ed., Graduate Texts in Mathematics, vol. 11, Springer-Verlag, New York-Berlin, 1978. MR 503901
- A. Zygmund, Trigonometric series. Vol. I, II, Cambridge University Press, Cambridge-New York-Melbourne, 1977. Reprinting of the 1968 version of the second edition with Volumes I and II bound together. MR 0617944
- Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
Additional Information
- Javad Mashreghi
- Affiliation: Department of Mathematics and Statistics, McGill University, Montreal, Canada H3A 2K6
- MR Author ID: 679575
- Email: mashregh@math.mcgill.ca
- 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
- © Copyright 2001 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 130 (2002), 683-688
- MSC (2000): Primary 30D20; Secondary 42A50
- DOI: https://doi.org/10.1090/S0002-9939-01-06335-3
- MathSciNet review: 1866020