Fourier method in the space of φB - distributions

Abstract

In our previous articles, we introduced and explored the notion of φB -distributions with values in the Banach space. This offers a new perspective on the theory of solvability of linear problems, which is important for solving partial differential equations, especially equations with deviating arguments. Here, we provide an overview of the theory of such distributions, propose a new approach to justify the use of the Fourier method for solving linear problems, and write out a correctly solvable problem for a system of partial differential equations with deviating arguments.