On generalized Takagi functions (Q581571)

Takagi's nondifferentiable function is \(\sum 2^{-n}\phi_ n(x)\), where \(\phi_ 1(x)=2-2x\) for \(\leq x\leq 1\), \(2x\) for \(0\leq x\leq\), and \(\phi_ n(x)=\phi (\phi_{n-1}(x)).\) The author studies properties of series of the form \(f(x)=\sum c_ n\phi_ n(x).\) For example, he obtains conditions for f to be nowhere differentiable, or absolutely continuous, or smooth. He also shows that \(\{\phi_ n(x)-\}\) is a multiplicative system, but not strongly multiplicative.