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

After recalling the notion of a complete metric space (Y,dY)(Y,dY) of measure-valued images over a base (or pixel) space X, we define a complete metric space (F,dF)(F,dF) of Fourier transforms of elements μ∈Yμ∈Y. We also show that a fractal transform T:Y→YT:Y→Y induces a mapping M on the space FF. The action of M on an element U∈FU∈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 FF: as expected, the fixed point View the MathML sourceU¯ of M is the Fourier transform of μ∈Y