Harmonic Analysis and Boundary Value Problems in the Complex Domain: (v. 65 Operator Theory: Advances and Applications)

The present book is a valuable continuation of the large cycle of the author's investigations on harmonic analysis in the complex domain. For certain sets of segments in the complex domain, the elegant and explicit apparatus of the biorthogonal Fourier type systems (basic systems in the Rieszian sense), is constructed by purely analytic methods of classical function theory. This is done using the remarkable asymptotic properties of the Mittag-Leffler type entire functions and new interpolation theorems for the Banach spaces of entire functions. It is especially noteworthy that at the same time the constructed basis systems are eigenfunctions for quite non-ordinary boundary value problems in the complex domain for differential equations of fractional order. Such boundary value problems, the solutions of which are carried through to a logical conclusion, i.e. up to the theorems of expansions into eigenfunctions, are considered in the book for the first time.

Table of Contents:
Part 1 Preliminary results. Integral transforms in the complex domain: Introduction; Some identities; Integral representations and asymptotic formulas; Distribution of zeros; Identities between some Mellin transforms; Fourier type transforms with Mittag-Leffler kernels; Some consequences; Notes. Part 2 Further results. Wiener-Paley type theorems: Introduction; Some simple generalizations of the first fundamental Wiener-Paley theorem; A general Wiener-Paley type theorem and some particular results; Two important cases of the general Wiener-Paley type theorem; Generalizations of the second fundamental Wiener-Paley theorem; Notes. Part 3 Some estimates in Banach spaces of analytic functions: Introduction; Some estimates in Hardy classes over a half-plane; Some estimates in weighted Hardy classes over a half-plane; Some estimates in Banach spaces of entire functions of exponential type; Notes. Part 4 Interpolation series expansions in spaces W of entire functions: Introduction; Lemmas on special Mittag-Leffler type functions; Two special interpolation series; Interpolation series expansions; Notes. Part 5 Fourier type basic systems in L2(0,U): Introduction; Biorthogonal systems of Mittag-Leffler type functions and their completeness in L2 (0, O); Fourier series type biorthogonal expansions in L2(0, o); Notes. Part 6 Interpolation expansions in the spaces Wpw s+1/2,a of entire functions: Introduction; The formulation of the main theorems; Auxilliary relations and lemmas; Further auxiliary results; Proofs of the main theorems; Notes. Part 7 Basic Fourier type systems in L2 spaces of odd-dimensional vector functions: Introduction; Some identities; Biorthogonal systems of odd-dimensional vector functions; Theorems on completeness and basis property; Notes. Part 8 Interpolation series expansions in the spaces Wpws,a of entire functions: Introduction; The formulation of the main interpolation theorem; Auxiliary relations and lemmas; Further auxiliary results; The proof of the main interpolation theorem; Notes. Part 9 Basic Fourier type systems in L2 spaces of even-dimensional vector functions: Introduction; Some identities; The construction of biorthogonal systems of even-dimensional vector functions; Theorems on completeness and basis property; Notes. Part 10 The simplest Cauchy type problems and the boundary value problems connected with them: Introduction; Riemann-Liouville fractional integrals and derivatives; A Cauchy type problem; The associated Cauchy type problem and the analog of Lagrange formula; Boundary value problems and eigenfunction expansions; Notes. Part 11 Cauchy type problems and boundary value problems in the complex domain (the case of odd segments): Introduction; Preliminaries; Cauchy type problems and boundary value problems containing the operators Ls+(Y2s+1(a)) in terms of Riesz bases; Notes. Part 12 Cauchy type problems and boundary value problems in the complex domain (the case of even segments): Introduction; Preliminaries;