Harmonic analysis and boundary value problems in the complex domain
type: Item
instance of: scholarly article
title: Harmonic analysis and boundary value problems in the complex domain
zbMATH Open document ID: 0798.43001
author: Mkhitar M. Djrbashian
published in: Operator Theory: Advances and Applications
publication date: 1993-12-14T00:00:00Z
review text: The Mittag-Leffler type entire function \(E_ \rho(z;\mu) = \sum^ \infty_{k = 0} z^ k/\Gamma(\mu + k/\rho)\) (\(\rho,\mu > 0\)) admits many remarkable asymptotic properties, namely functional equation, integral presentations, asymptotic formulas, good distribution of zeros, etc. \dots, which were detailly exposed in the author's previous book of 1986. In the book under review, the author summarizes these materials in chapters 1 and 2, extends them in chapter 3 and establishes in chapters 4, 6, 8 also some interpolation theorems for the Banach spaces of entire functions of order \(\rho = s - 1/2\) and \(\rho = s\), for a natural number \(s\). Based on these results, in the chapters 5, 7, 9 the author studies the harmonic analysis on certain sets of segments, coming out from the point \(z = 0\) of the Riemann surface \(G^ \infty\) of the function \(\text{Ln }z\). In particular he points out some Rieszian biorthogonal Fourier type systems. In chapters 10, 11, 12, these biorthogonal systems are represented as the systems of eigenfunctions and associated functions for certain non-ordinary boundary problems for differential operators of fractional order on suitable sets of the above indicated intervals. In general the book is a beautiful exposition of a large cycle of the author's research in exploiting the interrelation between harmonic analysis and complex analysis. It can also be recommended as a quick introduction to the subject.
reviewed by: Do Ngoc Diep
Mathematics Subject Classification ID: 43-02
Mathematics Subject Classification ID: 46E20
Mathematics Subject Classification ID: 33C80
Mathematics Subject Classification ID: 35K99
zbMATH DE Number: 481932
zbMATH Keywords: Mittag-Leffler type entire function
zbMATH Keywords: asymptotic properties
zbMATH Keywords: biorthogonal Fourier type systems
zbMATH Keywords: differential operators
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 481932