Volume III - Holomorphic Extensions

The idea igniting the study of the holomorphic extension of functions in this series of papers was the theory of complex angular momentum, which was proposed at the end of the fifties of the past century. In that theory relevant functions, like the scattering amplitude, had to be analyzed in extended complex domains as a consequence of the complexification of certain dynamical variables. In this collection of papers the extension of the domain of holomorphy has been exploited primarily for investigating the analyticity properties in complex domains of trigonometric, Fourier-Legendre and power series. An important example is the study of the holomorphy properties of certain Fourier series in complex domains which allows to solve the non-perturbative problem of recovering the thermal Green functions at real times from the corresponding Matsubara data at imaginary times. The associated analysis of the properties of functions at the boundary values of the extended domains yielded further insight into significant problems either physical, like the inverse scattering problem, and mathematical, e.g. the solution of classes of Cauchy integral equations, and moreover, within the context of the complex interpolation problem, led to an extended Shannon-type sampling theorem associated with classes of meromorphic functions.