Spread of values of a Cantor-type fractal continuous nonmonotone function
Using a certain representation (called \(Q_5^*\)-representation) for the numbers in \([0,1]\) that involves a quinary alphabet and an infinite stochastic matrix, the authors define an interval continuous function \(f\) whose description can be seen in the abstract of the paper under review; after revising some already known properties of \(f\) established in [\textit{M. V. Prats'ovytyi} and \textit{O. V. Svynchuk}, ``On one family of continuous nonmonotone singular functions of the Cantor-type with fractal properties'', Nauk. Chasopys. Nats. Ped. Univ. Dragomanova, Ser. 1, Fiz.-Mat. Nauk., No. 17, 37--48 (2015)] (among others, \(f\) is continuous, takes all values from \([0,1]\), does not have intervals of monotonicity except the intervals where it is constant), they characterize the bounded variation of \(f\) in terms of the convergence character of \(\sum_{n\geq 1}\varepsilon_n\), where the numbers \(\varepsilon_n\in[0,1]\) are those involucrated in the definition of \(f\). Also, the authors analyze the level sets of \(f\) and describe the images under \(f\) of Cantor-type sets. They connect in a natural way the \(Q_5^*\)-representation with random variables \(X\) and are able to obtain information of the corresponding distribution of the random variable \(Y=f(X)\) under additional hypothesis.
scientific article; zbMATH DE number 7139498