On the Conditions of Ensuring Uniform Almost Periodic Functions of the Class of Entire Functions

The study of almost periodic functions occupies an important place in functional analysis and the theory of differential equations, beginning with the classical works of H. Bohr, A. S. Besikovich and B. M. Levitan. Almost periodic functions, being a generalization of periodic functions, are characterized by the fact that they retain their structure under shifts, without being strictly periodic. On the other hand, entire functions are functions of a complex variable that are analytic in the entire complex plane. Their behavior, especially their growth and the location of their zeros, is studied in detail in the theory of functions of a complex variable. Of particular interest is the study of entire functions whose values on the real axis are almost periodic in the sense of Bohr. The question of under what conditions an entire function takes on values on the real axis that form a uniformly almost periodic function is a non-trivial problem at the intersection of function theory and spectral analysis. Such conditions can be formulated through the properties of the spectrum of the function, through the conditions on the coefficients of the Fourier series, and also through the growth properties of the function itself. These functions find application in spectral theory, quantum mechanics, oscillation theory, and other areas of mathematics and physics. In this section, we study the problems of approximation of functions <i>f(x)∈B </i>by entire functions of finite degree with arbitrary Fourier exponents. We establish necessary and sufficient conditions for functions <i>f(x)∈B </i>to belong to the class of entire functions of bounded degree.