On Certain Initial Techniques to Model Singular Functions
DOI:
https://doi.org/10.64229/1gamwr79Keywords:
Singular functions, Systems of functional equations, Compositions of functions, Expansions of real numbers, Probability distributionsAbstract
Nowadays many researches are devoted to pathological mathematical objects of real analysis such as the Moran and Cantor-like fractal sets, functions with complicated local structure, as well as their generalizations and various applications. For example, these applications include the development of fractal multiformalism, general fractal measures and dimensions, as well as Hewitt-Stromberg measures and homogeneous Moran measures, physical and economical modeling, etc. According to historical way of investigations in mathematics, it is important the question on the methodological tools used by classical mathematicians to advance the study of pathological functions. The present survey is devoted to examples and the main techniques for modeling mainly singular functions introduced before 2000 in papers indexed in Scopus. Since the later research in this topic must more explanations, the rest examples will be considered in next papers of the author. The main considered techniques to construct singular functions are following: using systems of functional equations; applying Markov chains and distribution function; using various expansions of arguments and values of a function; certain geometrical iteration procedures; using auxiliary relations for the geometric construction of the graph of a function; applications of auxiliary maps, compositions of functions, and iterated function systems.
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