Review of Methods, Applications and Publications on the Approximation of Piecewise Linear and Generalized Functions

Approximation of piecewise linear and generalized functions is an important and difficult problem. These functions are widely used in mathematical modeling of various processes and systems, such as: automatic control theory, electrical engineering, radio engineering, information theory and transmission of signals and images, equations of mathematical physics, oscillation theory, differential equations and many others. The widespread use of such functions is explained by their positive properties. For example, piecewise linear functions are characterized by a simple structure over segments. However, these features also have disadvantages. For example, in the case of using piecewise linear functions, solutions have to be built in segments. In this case, the problem of matching the obtained solutions at the boundaries of the segments arises, which leads to the complication of the research results. The use of generic functions has similar disadvantages. To eliminate shortcomings in practice, one resorts to the approximation of these functions. There are a large number of well-known methods for approximating piecewise linear and generalized functions. Recently, new methods for their approximation have been developed. In this study, an attempt was made to generalize and discuss the existing methods for approximating the considered functions. Particular emphasis is placed on the description of new approximation methods and their applications in various fields of science and technology. The publication-based review discusses the strengths and weaknesses of each method, compares them, and considers suitable application examples. The review will undoubtedly be interesting not only for mathematicians, but also for specialists and scientists working in various applied fields of research.