Singular functions with applications to fractal dimensions and generalized Takagi functions
The authors study the family of singular functions \(\{S_a\}\) -- depending on the real parameter \(a\) -- introduced by Césaro in 1906 and Hellinger in 1907, with the aid of a generalized dyadic system (recall that a singular function is a monotone increasing, continuous real function whose derivative is zero almost everywhere). In particular, following \textit{M. Hata} and \textit{M. Yamaguti} [Japan J. Appl. Math. 1, No. 1, 183--199 (1984; Zbl 0604.26004)], they study the connection between \(\{S_a\}\) and the class of generalized Takagi functions \(\{T_a\}\) -- a class of monotonic type on no-interval and almost everywhere differentiable functions tightly related to the classic continuous nowhere differentiable Takagi function. As an application to fractal dimension, they compute the Hausdorff dimension of certain subsets of the unit interval related to these functions.
scientific article; zbMATH DE number 6118579
scientific article