In the 1950s, the method of initial functions (MIF) was developed in the Soviet Union. It rapidly became popular among research scientists, civil engineers, and, later, among strength engineers engaged in the aerospace industry. Since the MIF is known comparatively little to the Western reader, a brief overview of the publications devoted to the formation and development of the MIF and its application to the solution of various engineering problems is given in the Introduction. In this paper, the MIF is considered in the space of Fourier transforms. This allows its use to be facilitated still further by reducing the solution of boundary value problems to simple algebraic transformations. The final solutions are represented either as improper Fourier integrals or as expansions into series in eigenfunctions of the boundary value problem, Papkovich–Fadle eigenfunctions. Using various examples, we show the basic techniques of working with the MIF in the space of Fourier transforms. In the final section, the method is applied to the solution of problems for a plane with displacement discontinuities. The solutions are obtained rapidly, easily, and, in contrast to the classical solution, without using the theory of functions of a complex variable.