Harmonic analysis and boundary value problems in the complex domain
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.
scientific article; zbMATH DE number 481932