Fourier transforms of measure-valued images, self-similarity and the inverse problem

After recalling the notion of a complete metric space (Y,dY) of measure-valued images over a base (or pixel) space X, we define a complete metric space (F,dF) of Fourier transforms of elements μ∈Y. We also show that a fractal transform T:Y→Y induces a mapping M on the space F. The action of M on an element U∈F is to produce a linear combination of frequency-expanded copies of M. Furthermore, if T is contractive in Y, then M is contractive on F: as expected, the fixed point U¯ of M is the Fourier transform of μ∈Y.