Entire functions. Exponential series. Textbook. (Tselye funktsii. Ryady ehksponent. Uchebnoe posobie).

*(Russian)*Zbl 0547.30003
Moskva: ”Nauka”. Glavnaya Redaktsiya Fiziko-Matematicheskoj Literatury. 176 p. R. 0.30 (1983).

This is a carefully written manual, reflecting a university course on the classical theory of entire functions of one complex variable. It is divided into two parts, the first dealing with the basic notions of order, type, growth, distribution of zeros, various infinite products, indicator diagram, Borel transform, completeness, various estimations. The second part is devoted to systems of exponentials, \((1)\quad\{\exp \lambda_ kz\}, k=1,2,...\), \(\lambda_ k\in {\mathbb{C}}\), \(|\lambda_ k|\nearrow \), and in particular to exponential series (Dirichlet series), \((2)\quad\sum a_ n\exp \lambda_ nz.\) These are studied by a well-known method through certain entire functions \(L(\lambda)\) of exponential type admitting the exponents \(\lambda_ k\) as zeros and systems biorthogonal to (1) defined through the function L. The simplest hypothesis is \(0<\lambda_ k\nearrow\infty,\quad\lim k/\lambda_ k\) exists, and this case is thoroughly studied. The applications are to finding the domain where the sum (2) is holomorphic, the singularities on the boundary, questions of completeness, estimations of finite sums (2), convolution equations, a version of spectral synthesis (subspaces invariant under differentiation).

In the final sections, the convergence to F of the exponential series of F is studied: \(F(z)\sim\sum a_ k\exp \lambda_ kz,\quad a_ k=(1/2\pi i)\int_{C}\psi_ k(t)F(t)dt,\) where \(\{\psi_ k\}\) is biorthogonal to \(\{\) exp \(\lambda {}_ kz\}\), C is a contour surrounding the closure of the conjugate diagram belonging to the entire function L of exponential type admitting the \(\lambda_ k\) as zeros.

In the final sections, the convergence to F of the exponential series of F is studied: \(F(z)\sim\sum a_ k\exp \lambda_ kz,\quad a_ k=(1/2\pi i)\int_{C}\psi_ k(t)F(t)dt,\) where \(\{\psi_ k\}\) is biorthogonal to \(\{\) exp \(\lambda {}_ kz\}\), C is a contour surrounding the closure of the conjugate diagram belonging to the entire function L of exponential type admitting the \(\lambda_ k\) as zeros.

Reviewer: E.J.Akutowicz

##### MSC:

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

30B60 | Completeness problems, closure of a system of functions of one complex variable |

30B50 | Dirichlet series, exponential series and other series in one complex variable |

30D20 | Entire functions of one complex variable, general theory |