The multifractal formalism for functions has been proved to be valid for a large class of selfsimilar functions. All the functions that have been studied are (or turned out to be) associated with a family of contractions which satisfies some separation conditions. In this paper, we extend the validity in the presence of overlaps involved by the well-known $n$-scale dilation family. Our method of proof is based on wavelet analysis and some interesting properties of this family.

A function f is self-similar if, modulo an error regular function g, it (i.e., f ) is invariant under specific transformations involving mainly dilations and translations (known as the piecewise linear dynamical systems). The multi-fractal formalism for functions has been proved to be valid for a large class of self-similar functions. For the computation of the spectrum of singularities for such functions, Gibbs measures were constructed. The authors have extended the validity for quasi-self-similar functions of the form (17).