Abstract
The article is dedicated to the issues of studying the approximation of functions by trigonometric polynomials with a spectrum from special sets. In this paper, these special sets are harmonic intervals. To study the approximation of functions over harmonic intervals, families of function classes have been created, designed as a subsidiary tool. These families of function classes are characterized through the best approximations of functions by trigonometric polynomials over such sets and are used in the research. For these families of function classes, their properties and the connection with classical Besov spaces are shown. The results of the study are presented in the form of theorems and lemmas. In carrying out the research presented in the article, the main apparatus for proving theorems are the fundamentals of approximation theory, the method of real interpolation of spaces, and the fundamentals of the theory of embedding classes of functions and functional spaces. The article is destined for mathematicians and can be used by researchers and specialists whose interests lie in the indicated areas of mathematics.