Boundedness of the discrete Hilbert transform in discrete Hölder spaces

Abstract

The Hilbert transform plays an important role in the theory and practice of signal processing operations in continuous system theory. The Hilbert transform was the motivation for the development of modern harmonic analysis. Its discrete version is also widely used in many areas of science and technology and plays an important role in digital signal processing. The essential motivation behind thinking about discrete transforms is that experimental data are most frequently not taken in a continuous manner but sampled at discrete time values. Since much of the data collected in both the physical sciences and engineering are discrete, the discrete Hilbert transform is a rather useful tool in these areas for the general analysis of this type of data. The Hilbert transform has been well studied on classical function spaces such as H¨older, Lebesgue, Morrey, etc. But its discrete version, which also has numerous applications, has not been fully studied in discrete analogues of these spaces. In this paper, we discuss the discrete Hilbert transform in discrete H¨older spaces and obtain its boundedness in these spaces.