The Takagi function and its generalization. (English) Zbl 0604.26004

Let \(\psi\) be the ”tent” map \(\psi (x)=1-| 2x-1|\) for \(x\in [0,1]\), and let \(E_ T\) be the Takagi’s class consisting of functions of the form \(\sum^{\infty}_{n=1}a_ n\psi^ n(x)\), where \(\sum a_ n\) is absolutely convergent and \(\psi^ n\) denotes the n-th iterate of \(\psi\). It is known that \(E_ T\) contains both singular and regular functions (i.e. \(a_ n=2^{-n}\) gives a nowhere differentiable map, while \(a_ n=4^{-n}\) gives the map \(g(x)=x(1-x)).\)
This interesting paper contains a characterization of \(E_ T\) using Schauder’s bases. The main part of the paper is devoted to the problem of existence of \(f\in E_ T\) with Schauder’s coefficients satisfying certain functional equations.
Reviewer: J.Smítal


26A30 Singular functions, Cantor functions, functions with other special properties
26A18 Iteration of real functions in one variable
26A27 Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives
39B12 Iteration theory, iterative and composite equations
Full Text: DOI


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