##
**The logarithmic integral. Vol. I.**
*(English)*
Zbl 0665.30038

Cambridge Studies in Advanced Mathematics, 12. Cambridge (UK) etc.: Cambridge University Press. xvi, 606 p. £60.00; $ 89.50 (1988).

The book under review consists of two volumes, the second of which has not yet appeared. The main subject of the book is the study of the integral
\[
\int^{\infty}_{-\infty}(\log M(t)/(1+t^ 2))dt
\]
in the case of the real line and
\[
\int^{\pi}_{-\pi}\log P(\vartheta)d\vartheta
\]
in the case of the unit cirlce. This integral plays an important role in many questions of mathematical analysis. The first volume consists of 8 chapters.

The first chapter is devoted to Jensen’s formula. Chapter II deals with Szegö’s theorem on weighted polynomial approximation. The second section of Chapter II concerns the pointwise approximate identity of the Poisson kernel.

Chapter III is called “Entire functions of exponential type”. It contains the Hadamard factorization theorem, Lindelöf’s characterization of zero sets of functions of exponential type, Phragmén-Lindelöf theorems, the Paley-Wiener theorem. The chapter concludes with an introduction to the condition \[ \int^{\infty}_{- \infty}(\log^+| f(x)| /(1+x^ 2))dx<\infty \] and Levinson’s theorem on the density of zeros of entire functions of Cartwright class.

Chapter IV concerns quasianalyticity. Carleman classes are introduced and Carleman’s criterion of quasianalyticity of Carleman classes is presented. The chapter also contains the Paley-Wiener construction of entire functions of small exponential type that decrease fairly rapidly along the real line.

Chapter V is called “The moment problem on the real line”. It deals with the Hamburger moment problem, i.e. given a sequence \(\{s_ n\}_{n\geq 0}\) of complex numbers, the problem is whether there exists a positive measure \(\mu\) on \({\mathbb{R}}\) with finite moments \(\int^{\infty}_{-\infty}| x|^ kd\mu (x)\) satisfying \[ s_ k=\int^{\infty}_{-\infty}x^ kd\mu (x),\quad k=0,1,2,3,.... \] The criterion of solvability is the positive definiteness of the corresponding Hankel matrix \(\{s_{n+k}\}_{n,k\geq 0}\). The rest of the chapter is devoted to the determinacy condition, i.e. the condition when the moment problem has exactly one solution. Several conditions are presented and the M. Riesz criterion is obtained.

Chapter VI concerns weighted approximation on the real line. Given a positive weight W on \({\mathbb{R}}\) satisfying \(x^ n/W(x)\to 0\) as \(x\to \pm \infty\), the problem of approximation by polynomials is considered in the space \({\mathcal C}_ w({\mathbb{R}})\) that consists of the continuous functions \(\phi\) on \({\mathbb{R}}\) satisfying \[ \phi (x)/W(x)\to \infty \quad as\quad x\to \pm \infty. \] Criterions due to Akhiezer and Mergelyan are given. An analogous problem is treated for approximation in \({\mathcal C}_ W({\mathbb{R}})\) by sums of imaginary exponentials with exponents from a finite interval. In the case when the set of such sums is not dense in \({\mathcal C}_ W({\mathbb{R}})\) a description due to L. de Branges of extremal unit measures orthogonal to this set is given. The same problems are also considered in the case of weighted \(L^ p\)-spaces. The last section of the chapter is devoted to comparison of weighted approximation by polynomials and linear combinations of exponentials.

Chapter VII is entitled “How small can the Fourier transform of a rapidly decreasing non-zero function be?” The subject of the chapter is focused on the following problem. Suppose that \(F\in L^ 1({\mathbb{R}})\) and satisfies \[ \int^{\infty}_{-\infty}\frac{1}{1+x^ 2}\log - (\int^{\infty}_{x}| F(t)| dt)dx=\infty. \] Then the problem is whether the Fourier transform \(\hat F\) can be small on some set if F does not vanish identically. The first result presented here is the Levinson theorem which claims that in this case \(\hat F\) cannot vanish on an interval of positive length. The second one is due to Beurling. It claims that \(\hat F\) cannot vanish even on a set of positive Lebesgue measure.

In the first section of the chapter the following Beurling gap theorem is presented:

Let \(\mu\) be a finite complex measure on \({\mathbb{R}}\) that vanish on disjoint intervals \((a_ n,b_ n)\) satisfying \[ \sum^{\infty}_{1}(a_ n-b_ n)/a_ n=\infty. \] Then the Fourier transform \({\hat \mu}\) cannot vanish on an interval of positive length. However an example of Kargaev given in the third section shows that here an interval cannot be replaced by a set of positive Lebesgue measure.

In the case of functions on the unit circle \({\mathbb{T}}\) the above results of Levinson and Beurling have recently been improved by Volberg who has proved the following beautiful result treated in the last section of the chapter. Let f be a continuous function on \({\mathbb{T}}\) whose Fourier coefficients satisfy \(| \hat f(n)| \leq e^{-M(n)}\) for \(n>0\) with M(n) sufficiently regular and increasing. If \[ \sum^{\infty}_{1}M(n)/n^ 2=\infty \quad then\quad \int^{\pi}_{- \pi}\log | f(\vartheta)| d\vartheta >-\infty \] unless \(f(\vartheta)\equiv 0.\)

The last chapter of the first volume is called “Persistance of the form \(dx/(1+x^ 2)\)”. It deals with estimation of harmonic measure on E, where E is a closed subset of \({\mathbb{R}}\). It is shown that under some natural conditions the harmonic measure behaves in the same way as the restriction to E of the harmonic measure \(dx/(1+x^ 2)\) on \({\mathbb{R}}\). In the first section the case when E has positive lower uniform density is considered. The second section concerns the limiting case when E is discrete. Namely, if \(E={\mathbb{Z}}\) and P is a polynomial such that \[ \sum^{\infty}_{-\infty}(1+n^ 2)^{-1}\log | P(n)| \leq \eta \] with \(\eta >0\) sufficiently small, then \(| P(z)| \leq K(z,\eta)\) with K dependent on z and \(\eta\), but not on P. The last section treats the case when E contains all points \(x\in {\mathbb{R}}\) of sufficiently large modulus. The volume ends with an addendum which is called “Improvement of Volberg’s theorem on the logarithmic integral. Work of Brennan, Borichev, Jöricke and Volberg”. It contains recent results of the authors mentioned in the title.

The first chapter is devoted to Jensen’s formula. Chapter II deals with Szegö’s theorem on weighted polynomial approximation. The second section of Chapter II concerns the pointwise approximate identity of the Poisson kernel.

Chapter III is called “Entire functions of exponential type”. It contains the Hadamard factorization theorem, Lindelöf’s characterization of zero sets of functions of exponential type, Phragmén-Lindelöf theorems, the Paley-Wiener theorem. The chapter concludes with an introduction to the condition \[ \int^{\infty}_{- \infty}(\log^+| f(x)| /(1+x^ 2))dx<\infty \] and Levinson’s theorem on the density of zeros of entire functions of Cartwright class.

Chapter IV concerns quasianalyticity. Carleman classes are introduced and Carleman’s criterion of quasianalyticity of Carleman classes is presented. The chapter also contains the Paley-Wiener construction of entire functions of small exponential type that decrease fairly rapidly along the real line.

Chapter V is called “The moment problem on the real line”. It deals with the Hamburger moment problem, i.e. given a sequence \(\{s_ n\}_{n\geq 0}\) of complex numbers, the problem is whether there exists a positive measure \(\mu\) on \({\mathbb{R}}\) with finite moments \(\int^{\infty}_{-\infty}| x|^ kd\mu (x)\) satisfying \[ s_ k=\int^{\infty}_{-\infty}x^ kd\mu (x),\quad k=0,1,2,3,.... \] The criterion of solvability is the positive definiteness of the corresponding Hankel matrix \(\{s_{n+k}\}_{n,k\geq 0}\). The rest of the chapter is devoted to the determinacy condition, i.e. the condition when the moment problem has exactly one solution. Several conditions are presented and the M. Riesz criterion is obtained.

Chapter VI concerns weighted approximation on the real line. Given a positive weight W on \({\mathbb{R}}\) satisfying \(x^ n/W(x)\to 0\) as \(x\to \pm \infty\), the problem of approximation by polynomials is considered in the space \({\mathcal C}_ w({\mathbb{R}})\) that consists of the continuous functions \(\phi\) on \({\mathbb{R}}\) satisfying \[ \phi (x)/W(x)\to \infty \quad as\quad x\to \pm \infty. \] Criterions due to Akhiezer and Mergelyan are given. An analogous problem is treated for approximation in \({\mathcal C}_ W({\mathbb{R}})\) by sums of imaginary exponentials with exponents from a finite interval. In the case when the set of such sums is not dense in \({\mathcal C}_ W({\mathbb{R}})\) a description due to L. de Branges of extremal unit measures orthogonal to this set is given. The same problems are also considered in the case of weighted \(L^ p\)-spaces. The last section of the chapter is devoted to comparison of weighted approximation by polynomials and linear combinations of exponentials.

Chapter VII is entitled “How small can the Fourier transform of a rapidly decreasing non-zero function be?” The subject of the chapter is focused on the following problem. Suppose that \(F\in L^ 1({\mathbb{R}})\) and satisfies \[ \int^{\infty}_{-\infty}\frac{1}{1+x^ 2}\log - (\int^{\infty}_{x}| F(t)| dt)dx=\infty. \] Then the problem is whether the Fourier transform \(\hat F\) can be small on some set if F does not vanish identically. The first result presented here is the Levinson theorem which claims that in this case \(\hat F\) cannot vanish on an interval of positive length. The second one is due to Beurling. It claims that \(\hat F\) cannot vanish even on a set of positive Lebesgue measure.

In the first section of the chapter the following Beurling gap theorem is presented:

Let \(\mu\) be a finite complex measure on \({\mathbb{R}}\) that vanish on disjoint intervals \((a_ n,b_ n)\) satisfying \[ \sum^{\infty}_{1}(a_ n-b_ n)/a_ n=\infty. \] Then the Fourier transform \({\hat \mu}\) cannot vanish on an interval of positive length. However an example of Kargaev given in the third section shows that here an interval cannot be replaced by a set of positive Lebesgue measure.

In the case of functions on the unit circle \({\mathbb{T}}\) the above results of Levinson and Beurling have recently been improved by Volberg who has proved the following beautiful result treated in the last section of the chapter. Let f be a continuous function on \({\mathbb{T}}\) whose Fourier coefficients satisfy \(| \hat f(n)| \leq e^{-M(n)}\) for \(n>0\) with M(n) sufficiently regular and increasing. If \[ \sum^{\infty}_{1}M(n)/n^ 2=\infty \quad then\quad \int^{\pi}_{- \pi}\log | f(\vartheta)| d\vartheta >-\infty \] unless \(f(\vartheta)\equiv 0.\)

The last chapter of the first volume is called “Persistance of the form \(dx/(1+x^ 2)\)”. It deals with estimation of harmonic measure on E, where E is a closed subset of \({\mathbb{R}}\). It is shown that under some natural conditions the harmonic measure behaves in the same way as the restriction to E of the harmonic measure \(dx/(1+x^ 2)\) on \({\mathbb{R}}\). In the first section the case when E has positive lower uniform density is considered. The second section concerns the limiting case when E is discrete. Namely, if \(E={\mathbb{Z}}\) and P is a polynomial such that \[ \sum^{\infty}_{-\infty}(1+n^ 2)^{-1}\log | P(n)| \leq \eta \] with \(\eta >0\) sufficiently small, then \(| P(z)| \leq K(z,\eta)\) with K dependent on z and \(\eta\), but not on P. The last section treats the case when E contains all points \(x\in {\mathbb{R}}\) of sufficiently large modulus. The volume ends with an addendum which is called “Improvement of Volberg’s theorem on the logarithmic integral. Work of Brennan, Borichev, Jöricke and Volberg”. It contains recent results of the authors mentioned in the title.

Reviewer: V.V.Peller

### MSC:

30D60 | Quasi-analytic and other classes of functions of one complex variable |

30-02 | Research exposition (monographs, survey articles) pertaining to functions of a complex variable |