This book treats the Hilbert transform. Applications of this transform, presented in the book, consist mainly of the inversion of the Cauchy integrals over an interval. The numerous applications to signal analysis are not discussed, the physical applications related to dispersion relations are only mentioned (a few lines on p. 89).
The book gives some material on distribution theory in Ch. 1. The presentation here is not self-contained: many results are referenced but not proved. In the literature one finds several books in which the distribution theory is presented in a self-contained way and in more detail.
In Ch. 2 the Riemann-Hilbert problem is discussed. This topic is developed in depth in the known books of N. Muskhelishvili and F. Gakhov. The author did not even mention the notion of index, which is the key notion in the theory. As a result, his presentation is somewhat formal and not very clear. For instance, it is not discussed if eq. (2.15) has a non-trivial solution and how the index of the coefficient in this equation influences the solvability of the equation.
In Chapters 3, 4 and 5, the definition and properties of the Hilbert transform on some classes of functions and distributions are discussed. Many results are cited without proofs. In particular, there are many references to the book by Titchmarsh. The definition of the Hilbert transform given by Gelfand and Shilov is discussed. The author writes (p. 121) that these authors made a wrong claim, but he does not explain what is the error and why the claim is wrong. On p. 122 the author writes that the test space consists of functions which may have discontinuities at the origin but their first derivatives are absolutely integrable. Since such functions must be absolutely continuous, the definition of the space of the test functions is not clear. The author studies the multidimensional Hilbert transform which is defined essentially as a product of the one-dimensional transforms.
In Ch. 6 the relation between the Fourier transform and the Hilbert transform is discussed, the representation of the operators which commute with dilations and translations is given, the Dirichlet problem (in the half-space) with distributional boundary values is discussed very briefly, and eigenvalues of the Hilbert transform are calculated.
In Ch. 7 the Hilbert transform of periodic functions is discussed. At the end of each chapter there are some exercises for the reader. Hints or answers are not provided. The exercises are for the most part routine. The book has a bibliography (112 entries) and an index. It does not contain a table of the Hilbert transforms. The standard table [e.g. G. Bateman and A. Erdelyi, Tables of integral transforms, New York (1954; Zbl 0058.34103 and Zbl 0055.36401)]contains 62 formulas, and to have a short table in the book on Hilbert transform is convenient for the readers.
There is no discussion of the numerical aspects: how does one compute the Hilbert transform of a function? This topic is certainly of interest to many, and they will be disappointed by not finding any discussion of it.
This book is a useful addition to the literature, since the existing literature on the Hilbert transform consists mainly of papers and parts of the books on integral transforms. It seems that no book devoted exclusively to the Hilbert transform has been published before.