Functions of bounded variation and their Fourier transforms
Fourier series and transform have been extensively studied for over hundred years. Convergence of Fourier series and integrals of different classes of functions, including functions of bounded variations, has also been studied for many years. So, a question naturally arises: Is there a need for a book on the Fourier transform of functions of bounded variations? \par The author begins by answering this question and explaining that his book may be considered as a second volume of a relatively recently published book [\textit{J. Appell} et al., Bounded variation and around. Berlin: de Gruyter (2014; Zbl 1282.26001)]. Since the publication of the latter some new results, as well as, different approaches to old problems, including generalizations to multi-dimensions, have appeared which necessitated compiling them in a book format. \par The book comprises two parts: Part I, One-dimensional case, and Part II, Multi-dimensional case, they are preceded by an introduction and followed by an afterward and the bibliography. Part I is made up of Chapters 1--4 and Part II of Chapters 5--9. In the introduction, the author states that instead of studying the Fourier transform of functions of bounded variations, the emphasis will be on their cosine and sine transforms given by \[ \hat{f}_c (x)=\int_0^\infty f(t) \cos (xt) dt, \] and \[ \hat{f}_s (x)=\int_0^\infty f(t) \sin (xt) dt, \] which may be combined in a single transform of the form \[ \hat{f}_\gamma (x)=\int_0^\infty f(t) \cos (xt-\pi \gamma /2) dt, \] where $\gamma = 0 \text{ or } 1$. \par Also reviewed in the introduction are some tools that will be used in later chapters, such as the Hilbert transform and the so-called balance operator defined by \[ B_{\phi} g(x)=\frac{1}{x^2}\int_0^\infty g(t/x) \phi (t)dt, \] which for $\phi(t)=\sin t $, yields $ \hat{g}_s (x)=x B_{\sin x}g(x). $ \par One of the author's goals is to extend some of the one-dimensional results to higher dimensions. Because of the variety of notions and definitions of multidimensional bounded variation, there is a variety of multidimensional Hardy spaces and Hilbert transforms. The author paves the way to his forthcoming work by briefly discussing his approach and explaining why he has preferred certain definitions over others. \par Chapter One presents the machinery needed for forthcoming chapters by recalling standard definitions and theorems, such as functions of bounded variation, absolutely continuous functions, Lebesgue theorem, Fourier, Hilbert, and Hilbert-Stieltjes transforms, Hardy spaces, atomic decomposition, the Paley-Wiener theorem, and finally the chapter is concluded with a discussion of the balance integral operator and its properties. It should be noted that the author defines the Hardy space $H^1=H(\mathbb{R})$ in terms of the integrability of the Hilbert transform. More precisely, the real Hardy space $H^1$ is the subspace of $L^1(\mathbb{R})$ which consists of all functions with integrable Hilbert transform, a definition that plays an important role in the following chapters. It is no surprise that Chapter One is the longest chapter in the book. \par Chapter Two is concerned with the integrability of the Fourier transforms of certain classes of functions, such as functions of bounded variations and convex functions. To motivate the reader, the author begins the chapter by giving three examples to point out that the cosine and sine transforms may have different integrability conditions. The first two examples are: \[ f(t)= t^{-1/2},\quad g(t)=e^{-t}, \] whose cosine and sine transforms are given by \[ \hat{f}_c (x)=\hat{f}_s (x) =\sqrt{\pi/2 x}, \] \[ \hat{g}_c (x)=\frac{1}{1+x^2} , \quad \hat{g}_s (x)=\frac{x}{1+x^2}. \] While $\hat{f}_c, \hat{f}_s$ are integrable over $(0,1)$ they are not integrable over $(0, \infty)$. On the other hand $\hat{g}_c$ is integrable over $(0, \infty)$ but $\hat{g}_s$ is integrable only over finite intervals. The third example is $ h(t)=1/t$, whose sine transform is $\hat{h}_s (x)=\pi/2 $, while its cosine transform $ \hat{h}_c (x)$ does not exist. \par The rest of the chapter is devoted to theorems on the integrability and the asymptotic behavior of the cosine and sine transforms of different classes of functions of bounded variations, in particular to those functions whose derivatives belong to subspaces of Hardy spaces. \par Chapter Three, which is based on results obtained previously by the same author, focuses on the asymptotic formula for the sine Fourier transform, in particular on functions $f\in BV_0[0, \infty)\cap LAC (0.\infty), $ where $BV_0[0, \infty)$ is the space of functions of bounded variations on $[0, \infty)$ that vanish at infinity and $ LAC (0.\infty)$ is the space of absolutely continuous functions on $(0, \infty).$ Another function space $Q$ is introduced as \[ Q=\left\{ g: g \in L^2(\mathbb R), \int_{\mathbb R} \frac{|\hat{g}(x)|}{|x|}dx < \infty\right\}. \] A number of results concerning the sine and cosine Fourier transforms of functions whose derivatives belong to $Q$ are obtained. \par Chapter Four serves as a bridge to higher dimension generalizations. It collects results obtained by the author and by M. R. Trigub on the Fourier transform of convex functions and on the use of the balance operator to find the asymptotic behavior of the cosine and sine transforms of functions of bounded variations and convex functions. A special result concerning the sine transform of absolutely continuous functions is derived. \par Part II deals with the multidimensional case. It aims at extending some of the results in Part I to higher dimensions. Many of the results obtained in two dimensions appear to be exactly the same as those in higher dimensions except for different notation. However, there are results in two or three dimensions that appear to be different. \par Chapter Five introduces new notation and definitions. Of particular importance is the $n$-dimensional vector $\mathbf{\eta}=(\eta_1, \dots ,\eta_n) $ with entries either zeros or ones only. Because of the different degrees of freedom in the multidimensional case, there is a number of definitions of bounded variations and absolute continuity in several variables. The focus here is on Vitali's, Hardy's, and Tonelli's variations. The chapter is concluded with a review of the $n$-dimensional Fourier transform and Poisson summation formula. \par Chapter Six extends some of the one-dimensional results to higher dimensions. As was done in the one-dimensional case, the Fourier cosine and sine transforms are combined together in the formula \[ \hat{f}_\eta (x)= \int_{\mathbb{R}_+^n}\left( \prod_{i: \eta_i=1} \cos x_i u_i \prod_{i: \eta_i=0} \sin x_i u_i \right) du. \] If $\eta =1$, we have purely cosine transform and if $\eta =0$, we have purely sine transform, otherwise we have mixed transforms. A number of results on the integrability of the Fourier transform of classes of functions are obtained. \par Sharper and more precise results for locally absolutely continuous functions with bounded Hardy variation in multidimensions are introduced in Chapter 7. The focus is on Fourier transform of the form \[ \hat{f}_\gamma (x) =\int_{\mathbb R^n_+ } f(u) \left( \prod_{j=1}^n \cos \left(x_ju_j - \pi \gamma_j /2 \right) \right)du, \] where $\gamma =( \gamma_1, \cdots , \gamma_n )$ with $\gamma_j= 0 \text{ or } 1$. When all the $\gamma_j =0 $, we obtain the cosine transform and when all the $\gamma_j =1$ we obtain the sine transform. To study integrability conditions for this transform, the standard notion of convexity appears to be inadequate and as a result a stronger notion called robust convexity is introduced and analogous results to those in Part I are derived. \par Chapter Eight examines the connection between trigonometric series and Fourier integrals of functions of bounded variation. For a one-dimensional function of bounded variation vanishing at infinity, the following relation holds \[ \sup_{|x|\leq \pi}\left| \int_{\mathbb{R}} f(t) e^{ixt} dt- \sum_{k\in \mathbb{Z}} f(k) e^{ikx} \right| \leq \left\|f \right\|_{BV}, \] where $f(k)$ are the Fourier coefficients of $f$. The main goal of the chapter is to generalize this result to higher dimensions. Employing a number of preliminary results, which the author calls bridges, he obtains the following important theorem: Let $f$ be of Hardy variation and vanish at infinity along with all its derivatives $D^\eta f$, except for $\eta=1$, where \[ D^\eta f(x)=\left( \prod_{j: \eta_j =1}\frac{\partial}{\partial x_j} \right) f(x). \] If $f$ and all its derivatives are absolutely continuous, then \[ \sum_{k\in \mathbb{Z}^n} f(k) e^{i \langle x, k \rangle} =\int_{\mathbb R^n}f(u) e^{i \langle x, u \rangle}du +\Phi(x), \] where $\Phi$ is an integrable function. The chapter is concluded with a generalized Poisson summation formula and then followed by Chapter Nine which discusses the integrability of the Fourier transform of radial functions. \par There are 211 items listed in the bibliography of which 33 are the author's own publications. The author's writing style is a mix of formal mathematics and occasionally casual language with metaphors and interesting quotes such as ``There is a continent of Fourier transforms in which countries of the Fourier transform civilization are located, the country of functions of bounded variation among them.'' But my favorite one is ``The best teachers are those who show you where to look, but don't tell you what to see.''
scientific article; zbMATH DE number 7024210