Real Analysis Exchange

On Exceptional Sets of the Hilbert Transform

Grigori A. Karagulyan

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Abstract

We prove several theorems concerning the exceptional sets of the Hilbert transform on the real line. In particular, it is proved that any null set is an exceptional set for the Hilbert transform of an indicator function. The paper also provides a real variable approach to the Kahane-Katsnelson theorem on divergence of Fourier series.

Article information

Source
Real Anal. Exchange, Volume 42, Number 2 (2017), 311-328.

Dates
First available in Project Euclid: 10 May 2018

Permanent link to this document
https://projecteuclid.org/euclid.rae/1525917690

Digital Object Identifier
doi:10.14321/realanalexch.42.2.0311

Mathematical Reviews number (MathSciNet)
MR3721804

Zentralblatt MATH identifier
06870332

Subjects
Primary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B05: Fourier series and coefficients

Keywords
Hilbert transform exceptional null set divergent Fourier series

Citation

Karagulyan, Grigori A. On Exceptional Sets of the Hilbert Transform. Real Anal. Exchange 42 (2017), no. 2, 311--328. doi:10.14321/realanalexch.42.2.0311. https://projecteuclid.org/euclid.rae/1525917690


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