Abstract
In this article we obtain an explicit formula for the Hilbert transform of \(\log |f|,\) for the function f in Nevanlinna class having continuous extension to the real line. This family is the largest possible for which such a formula for the Hilbert transform of \(\log |f|,\) can be obtained. The formula is very general and implies several previously known results.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bhardwaj, A.K., Chattopadhyay, Arup, Mashreghi, J., Srivastava R. K.: Hilbert Transform in the Cartwright-de Branges space, Operator Theory: Advances and Applications. Springer Nature. p. 14, to appear
Conway, J.: Functions of One Complex Variable, 2nd edn. Springer-Verlag, Berlin (1978)
D’yakonov, K.M.: Entire functions of exponential type and model subspaces in \(H^p\), (Russian) Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 190 (1991), Issled. po Lineĭn. Oper. i Teor. Funktsiĭ. 19, 81–100, 186; translation in J. Math. Sci. 71 (1994), no. 1, 2222–2233
Makarov, N., Poltoratski, A.: Meromorphic inner functions, Perspectives in Analysis, Toeplitz kernels and the Uncertainty Principle, Springer, pp. 185–252. Berlin (2005)
Makarov, N., Poltoratski, A.: Beurling-Malliavin theory for Toeplitz kernels. Invent. Math. 180(3), 443–480 (2010)
Mashreghi, J.: Representation Theorems in Hardy Spaces, London Mathematical Society Student Texts Series 74, Cambridge University Press, Cambridge (2009)
Mashreghi, J.: Hilbert transform of \(\log |f|\). Proc. Amer. Math. Soc. 130(3), 683–688 (2002)
Poltoratski, A.: Toeplitz approach to problems of the uncertainty principle. in: Conference Board of the Mathematical Sciences, vol. 121, American Mathematical Soc., (2015)
Acknowledgements
We acknowledge the support provided by IIT Guwahati (Government of India), SERB-DST (Government of India), and NSERC (Natural Sciences and Engineering Research Council of Canada).
Author information
Authors and Affiliations
Contributions
The contribution of each author is almosat equally proportional.
Corresponding author
Ethics declarations
Conflict of interest
The authors declare no Conflict of interest.
Additional information
Communicated by Hari Bercovici.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Bhardwaj, A.K., Mashreghi, J. & Srivastava, R.K. Hilbert Transform, Nevanlinna Class and Toeplitz Kernels. Complex Anal. Oper. Theory 18, 78 (2024). https://doi.org/10.1007/s11785-024-01521-5
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11785-024-01521-5