##
**The logarithmic integral. II.**
*(English)*
Zbl 0791.30020

Cambridge Studies in Advanced Mathematics. 21. Cambridge: Cambridge University Press. xxvi, 574 p. (1992).

For part I, see the review in (1988; Zbl 0665.30038).

This is a continuation of the first volume. Chapter IX demonstrates abilities of Jensen’s formula. It is used to prove two main results of the chapter: Pólya’s gap theorem and a lower bound for the completeness radius for a system of imaginary exponentials (i.e. the supremum of the lengths of intervals \(I\) such that the system of exponentials is dense on \(C(I))\). Those applications of Jensen’s formula are due to J.-P. Kahane. Chapter X is devoted to a discussion of “multiplier theorems”. A multiplier theorem in this book means the following. Let \(W\) be a positive function on \(\mathbb{R}\) such that \(W(t) \geq 1\), \(t \in \mathbb{R}\). The problem is under which conditions there exist a nonzero entire function \(\varphi\) of exponential type such that the product \(\varphi W\) is bounded on \(\mathbb{R}\) (or belongs to some \(L^ p\) class). There is an obvious necessary condition for that: \[ \int_ \mathbb{R} {\log W(t) \over 1+t^ 2} dt<\infty, \] which is not sufficient. The following deep result is due to Beurling and Malliavin: if \(F\) is an entire function of exponential type, then \(| F|+1\) admit multipliers if and only if \[ \int_ \mathbb{R} {\log^ +| F(t) | \over 1+t^ 2} dt<\infty. \] The proof of this theorem is given in Chapter XI, in Chapter X several applications of multiplier theorems are given. In particular it is shown that the lower bound for the completeness radius associated with a system of imaginary exponentials is also an upper bound (this result is also due to Beurling and Malliavin). Another result due to Beurling and Malliavin given in Chapter 10 is the evaluation of the completeness radius associated with a system of real exponentials. One more application concerns weighted quadratic estimates for harmonic conjugation. The last section of the chapter is devoted to the study of the space \(L^ \infty/H^ \infty\). Chapter XI is devoted to the proof of multiplier theorems. In particular a proof of the Beurling-Malliavin multiplier theorem stated in Chapter X is given.

This is a continuation of the first volume. Chapter IX demonstrates abilities of Jensen’s formula. It is used to prove two main results of the chapter: Pólya’s gap theorem and a lower bound for the completeness radius for a system of imaginary exponentials (i.e. the supremum of the lengths of intervals \(I\) such that the system of exponentials is dense on \(C(I))\). Those applications of Jensen’s formula are due to J.-P. Kahane. Chapter X is devoted to a discussion of “multiplier theorems”. A multiplier theorem in this book means the following. Let \(W\) be a positive function on \(\mathbb{R}\) such that \(W(t) \geq 1\), \(t \in \mathbb{R}\). The problem is under which conditions there exist a nonzero entire function \(\varphi\) of exponential type such that the product \(\varphi W\) is bounded on \(\mathbb{R}\) (or belongs to some \(L^ p\) class). There is an obvious necessary condition for that: \[ \int_ \mathbb{R} {\log W(t) \over 1+t^ 2} dt<\infty, \] which is not sufficient. The following deep result is due to Beurling and Malliavin: if \(F\) is an entire function of exponential type, then \(| F|+1\) admit multipliers if and only if \[ \int_ \mathbb{R} {\log^ +| F(t) | \over 1+t^ 2} dt<\infty. \] The proof of this theorem is given in Chapter XI, in Chapter X several applications of multiplier theorems are given. In particular it is shown that the lower bound for the completeness radius associated with a system of imaginary exponentials is also an upper bound (this result is also due to Beurling and Malliavin). Another result due to Beurling and Malliavin given in Chapter 10 is the evaluation of the completeness radius associated with a system of real exponentials. One more application concerns weighted quadratic estimates for harmonic conjugation. The last section of the chapter is devoted to the study of the space \(L^ \infty/H^ \infty\). Chapter XI is devoted to the proof of multiplier theorems. In particular a proof of the Beurling-Malliavin multiplier theorem stated in Chapter X is given.

Reviewer: V.V.Peller (Manhattan)

### MSC:

30D15 | Special classes of entire functions of one complex variable and growth estimates |

30D55 | \(H^p\)-classes (MSC2000) |

42C30 | Completeness of sets of functions in nontrigonometric harmonic analysis |