Definition of the Hilbert transform in En

In this chapter the elementary properties of the n-dimensional Hilbert transform are discussed. Basic aspects of the Calderón–Zygmund theory of singular integral operators in the n-dimensional Euclidean space, En, are also considered.

Applications of Hilbert transforms in En for n ≥ 2 are significantly less numerous than the one-dimensional case; however, they do arise in important areas. These include problems in nonlinear optics that focus on deriving dispersion relations and sum rules for the nonlinear susceptibility. The publications of Smet and Smet (1974), Nieto-Vesperinas (1980), Peiponen, (1987b), 1988), Bassani and Scandolo (1991, 1992), and Peiponen, Vartiainen, and Asakura (1999), will give the reader a sense of the advances in this field. There are applications in signal processing (see Bose and Prabhu (1979), Zhu, Peyrin, and Goutte (1990), and Reddy et al. (1991a, 1991b)), and in spectroscopy (see Peiponen, Vartiainen, and Tsuboi (1990)). In scattering theory, the double dispersion relations, frequently referred to as the Mandelstam representation, express the scattering amplitude as a double iterated dispersion relation in the energy and momentum transfer variables; see Roman (1965) and Nussenzveig (1972). Some of these applications are touched upon in later chapters.

To proceed, some preliminary definitions are required. Let x denote the n-tuple {x1, x2, x3, …, xn}, and let s denote the n-tuple {s1, s2, s3, …, sn}. It is quite common in the literature to represent multi-dimensional integration factors by ds (or some similar variable), where the context is meant to signify an n-dimensional integration factor ds1ds2ds3 … dsn.

Introduction

The terminology “discrete Hilbert transform” arises in two distinct contexts. The first occurrence is in the study of certain types of series with a denominator of the form n – m, where m is the summation index and n ∈ ℤ. The second area is in signal processing in engineering problems. The first part of this chapter is devoted to a discussion of the discrete Hilbert transform as it is commonly employed in engineering applications, and to some related topics including the discrete Fourier transform. The focus of the second part of the chapter is a discussion of the analog of the Hilbert transform on ℝ for a discrete series. Two related definitions are considered. The key inequality for the discrete Hilbert transform, called Hilbert's inequality, is established and a generalization given by M. Riesz is discussed. 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 discrete Fourier transform

The discrete Fourier transform is commonly denoted in the scientific literature by the acronym DFT. The ideas that evolve for this transform provide a foundation for the development of the discrete Hilbert transform. 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. This is the fundamental feature of digital signal processing.

Introduction

Principal value integrals arise in a wide variety of applications, and in many cases it is not possible to evaluate such integrals in a simple closed analytic form. Consequently, there has been a significant investment of research effort devoted to the numerical evaluation of principal value integrals. The expression “numerical quadrature”, or frequently just the term “quadrature”, is used synonymously with numerical integration.

Some of the numerical integration approaches that have been developed for principal value integrals are outlined in this chapter. These range from rather simple schemes, which are sometimes quite effective, to approaches that yield fairly precise results and can be implemented in a high–speed calculation. Methods that are discussed include Maclaurin's formula, the trapezoidal rule, Simpson's formula, specialized Gaussian quadrature methods, and techniques involving Fourier transforms, including the fast Fourier transform, Fourier allied integral approaches, and methods based on conjugate Fourier series. Since a number of principal value integral problems arise in the context of transforming experimental data, some attention is devoted to the discretized nature of the data and how this can be handled.

Even if an analytic solution can be found for a particular principal value integral, numerical methods can be employed as a very useful check on the closed form result.

Some elementary transformations for Cauchy principal value integrals

Two straightforward, but potentially very useful, transformations that may be employed to simplify the evaluation of the Hilbert transform are discussed in this section.

Introduction

The focus of this chapter is the derivation of dispersion relations arising in magnetooptical applications and for natural optical activity. A further topic considered is the development of sum rules for some of the quantities that are experimentally accessible. These include the optical rotatory dispersion (commonly abbreviated as ORD in the literature), circular dichroism (CD), the magnetic analogs, the Faraday dispersion, and magnetic circular dichroism (MCD). These terms will be defined shortly.

A number of the ideas that are employed are very similar to those developed in the preceding two chapters. There are, however, some major differences that arise for the cases of magneto-optical and natural optical activity. The dispersion relations for magneto-optical and natural optical activity have a different structure compared with what was found for the typical case of other optical properties discussed in Chapters 19 and 20. The difference arises primarily from the form of the crossing symmetry relations for the refractive indices and absorption coefficients for left and right circularly polarized light. The reason for the absence of simple dispersion relations for the individual modes corresponding to the refractive indices for left and right polarization, and the corresponding absorption coefficients for right and left circularly polarized light, is also discussed.

My objective in this book is to present an elementary introduction to the theory of the Hilbert transform and a selection of applications where this transform is applied. The treatment is directed primarily at mathematically well prepared upper division undergraduates in physics and related sciences, as well as engineering, and first–year graduate students in these areas. Undergraduate students with a major in applied mathematics will find material of interest in this work.

I have attempted to make the treatment self–contained. To that end, I have collected a number of topics for review in Chapter 2. A reader with a good undergraduate mathematics background could possibly skip over much of this chapter. For others, it might serve as a highly condensed review of material used later in the text. The principal background mathematics assumed of the reader is a solid foundation in basic calculus, including introductory differential equations, a course in linear and abstract algebra, some exposure to operator theory basics, and an introductory knowledge of complex variables. Readers with a few deficiencies in these areas will find a number of recommendations for further reading at the end of Chapter 2. Some of the applications discussed require the reader to be familiar with basic electrodynamics.

My objective in this book is to present an elementary introduction to the theory of the Hilbert transform and a selection of applications where this transform is applied. The treatment is directed primarily at mathematically well prepared upper division undergraduates in physics and related sciences, as well as engineering, and first-year graduate students in these areas. Undergraduate students with a major in applied mathematics will find material of interest in this work.

I have attempted to make the treatment self-contained. To that end, I have collected a number of topics for review in Chapter 2. A reader with a good undergraduate mathematics background could possibly skip over much of this chapter. For others, it might serve as a highly condensed review of material used later in the text. The principal background mathematics assumed of the reader is a solid foundation in basic calculus, including introductory differential equations, a course in linear and abstract algebra, some exposure to operator theory basics, and an introductory knowledge of complex variables. Readers with a few deficiencies in these areas will find a number of recommendations for further reading at the end of Chapter 2. Some of the applications discussed require the reader to be familiar with basic electrodynamics.