Initial-boundary value problems for linear PDEs: the analyticity approach

To study initial-boundary value problems for linear PDEs we have recently proposed two alternative approaches in Fourier space: the "analyticity appoach" and the "elimination by restriction approach". In this paper we present the analyticity appoach and we illustrate its power in studying the well-posedness of initial - boundary value problems for second and third order evolutionary PDEs, and in constructing their solution. We also show the connection between the analyticity approach and the elimination by restriction approach in the study of the Dirichelet and Neumann problems for the Schroedinger equation in the n-dimensional quadrant.