The book offers a wonderful overview, very well structured and complete, on the theory of harmonic analysis developed in the context of the Vilenkin-Fourier series.
Therefore, let \(\mathcal{N}_+\) be the set of the positive integers and let \(m=(m_0,m_1, \dots)\) be a sequence of positive integers not less than two. Let us consider the additive group of integers of modulo \(m_k\) which is denoted by \[ \mathcal{Z}_{m_k}=\{0,1,\dots m_k-1\}. \] Then, the group \(G_m\) is defined as the complete direct product of the groups \(\mathcal{Z}_{m_i}\) with the product of the discrets topologies of \(\mathcal{Z}_{m_j}'s\). The direct product \(\mu\) of the measures \(\mu_k(\{j\})=1/m_k\) with \(j\in \mathcal{Z}_{m_k}\) is the Haar measure on \(G_m\), where \(\mu(G_m)=1\). The elements of \(G_m\) are given by \(x=(x_0,x_1,\dots)\) with \(x_j\in \mathcal{Z}_{m_j}\).
Thus, the Vilekin system \(\psi=(\psi_n)_{n\in\mathbb{N}}\) on \(G_m\) is defined as \[ \psi_n(x)=\prod_{k=0}^\infty r_k^{n_k}(x), \ \ \ n\in\mathbb{N}, \] where \(r_k\) is the generalized Rademacher function.
In this context, the authors introduce Fourier-Vilekin coefficients \(C_j\) and Fourier-Vilekin series \(T(x)\) respectively as, \[ C_j:=\hat{f}(j)= \int_{G_m}f\bar{\psi}_j d\mu,\text{ and } T(x)=\sum_{j=0}^\infty C_j\psi_j, \] where \(f\in L^1(G_m)\) and \(j=0,1,2\dots\).
So, the book consists of ten chapters that can be basically divided into two thematic blocks: the first part, which covers the first five chapters, deals with a systematic study of the concepts developed for series Fourier-Vilenkin, such as summability methods, approximations of the identity, modulus of continuity, conditional expectation operators and the study of maximal operators, among others. This first block begins with the introduction of Vilekin groups and concludes with the notion of martingale Hardy spaces associated with Vilekin-Fourier expansions and the study of strong convergence and interpolation techniques between Lebesgue spaces, Hardy spaces, and Lorentz spaces. The second thematic block comprises from chapters six to chapter nine, where a systematic study of partial sums, maximal operators, Vilekin-Fejer means, Riesz and Nörlund Logarithmic means, T-means, and atomic decomposition is carried out, but now, in the framework of martingale Hardy spaces. Noteworthy that chapter nine deals with the extension of most of the results obtained in the previous chapters, but now considering variable Lebesgue spaces and variable martingale Hardy spaces. Finally, they close with an appendix that constitutes chapter ten, where they develop convergence results regarding the partial sums and the Césaro sums, by considering estimates of Walsh-Fejer kernels, modulus of continuity in \(H_p\), and maximal operators of Kaczmarz-Fejer-means but associated with expansions in Walsh systems and Walsh-Kaczmarz systems.
In the words of the authors themselves: “the aim of this book is to discuss, develop and apply the newest developments of this fascinating theory connected to modern harmonic analysis”. Thus, each chapter begins with a broad introduction rich in bibliographical citations that allow the reader to locate himself in the historical context of the topics to be addressed, and at the end from it the authors include an important and interesting list of open problems proposed with a view to being resolved in future research.