Entire functions. Exponential series. Textbook. (Tselye funktsii. Ryady ehksponent. Uchebnoe posobie). (Russian) Zbl 0547.30003

Leont’ev, A. F.

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.

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

Keywords:

entire functions of exponential type; systems of exponential functions; convergence of exponential series; interpolation; order; indicator diagram; Borel transform; completeness; Dirichlet series