We present a solution to a functional equation by means of the construction of a contractive operator on some functional space. This solutions presents a kind of self-similarity and enables us to generalize the model introduced by Cabrelli et al. in [CFMV92] allowing a much greater flexibility. In particular dilation equations of the type f(x) = P c k f(2x \Gamma k) fit into this model, and hence we can construct a multiresolution analysis in the sense of Mallat and Meyer. On the other hand, this "generalized self-similarity" notion provides us with a method for the construction of an operator whose fixed point is close to a given target
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