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Complex analysis and convolution equations. (Russian) Zbl 0706.46031

Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Fundam. Napravleniya 54, 5-111 (1989).
This paper is a fairly comprehensive survey on the most important developments which have taken place in the field of mean periodicity in the last ten years. The theory of convolution equations (surjectivity of convolution operators, singularity of solutions, integral representation of solutions, etc.) is approached from a unified point of view, which enables the authors to cover many (apparently disconnected) topics in multivariable complex analysis.
The first chapter of this work provides a fairly self-contained introduction to the theory of convolution equations and to the related problems in interpolation in \({\mathbb{C}}^ n\). The authors then explore the developments in the field of integral representation formulas, which have provided (since the work of Henkin, Ramirez, et al.) one of the most powerful new tools in complex variables. After touching upon topics in integral geometry (using the so called Pompeiu problem as a thread), and in the theory of multidimensional residues, the authors conclude their work with a chapter devoted to the description of the exciting field of algebraic analysis which has recently started to be employed for the solutions of many problems which bear substantial connections with the work under review. Quite recently, for example, the authors and Kawai have found new ways to exploit the techniques described in this work to attack outstanding problems in the theory of thetanull values (see the forthcoming paper by Berenstein, Kawai and Struppa).
Reviewer: D.Struppa

MSC:

46F15 Hyperfunctions, analytic functionals
44A35 Convolution as an integral transform
45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
47A50 Equations and inequalities involving linear operators, with vector unknowns
30E05 Moment problems and interpolation problems in the complex plane