De Rham's singular function, two-scale difference equations and Appell polynomials
The de Rham function, which is the main subject of this paper, is the (up to a multiplicative constant) general unbounded solution of the pair of functional equations \(\phi(t/2)=a\phi(t), \phi((1+t)/2)=a+(1-a)\phi(t) (t\in[0,1])\). For \(a\in ]0,1] \setminus \{1/2\}\) it is continuous and strictly monotonic but its derivative is almost everywhere 0. The authors offer further properties of this function, in particular its Hölder-continuity, and connections to Appell polynomials, to difference and integral-functional equations, and to wavelet theory.
scientific article; zbMATH DE number 1498576