On associated homogeneous Gelfand functions

The paper proposes refined definitions for associated homogeneous functions (AHFs) of real variables, which are of great practical importance for a wide range of problems. I. M. Gelfand and Z. Ya. Shapiro were the first in 1955 to introduce AHFs into scientific use. However, the possibilities of using these functions in various applications have not been exhausted to this day. The proposed definitions inherit the basic idea of the original paper to define chains of new functions using the recurrent linear functional relations, where some homogeneous Euler function is the starting point. This makes it possible to apply the corresponding results not only for differentiable and continuous functions, but also for discontinuous functions, including discontinuous ones at all points. The possibility of constructing a detailed consistent theory of AHFs of real variables, defined by a chain of linear recurrent functional relations of a general form, is shown. The basic theorems are formulated and proven. Further ways of generalizing the functions under consideration are discussed.