Approximation of the Hilbert Transform in Hölder Spaces

Year 2024, Volume: 14 Issue: 2, 88 - 98, 31.07.2024

Rashid Aliev , Lale Sh. Alizade

Abstract

The Hilbert transform plays an important role in the theory and practice
of signal processing operations in continuous system theory because of its relevance to
such problems as envelope detection and demodulation, as well as its use in relating the
real and imaginary components, and the magnitude and phase components of spectra.
The Hilbert transform is a multiplier operator and is widely used in the theory of Fourier
transforms. It is also the main part of the theory of singular integral equations on the real
line. Therefore, approximations of Hilbert transform are of great interest. Many papers
have dealt with the numerical approximation of singular integrals in case of bounded
intervals. On the other hand, the literature concerning the numerical integration on
unbounded intervals is much sparser than the one on bounded intervals. There is very
little literature concerning the case of Hilbert transform. This article is dedicated to
the approximation of Hilbert transform in H¨older spaces by the operators introduced by
V.R.Kress and E.Mortensen to approximate the Hilbert transform of analytic functions
in a strip.

Keywords

Hilbert transform, singular integral, approximation, H¨older spaces

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