We define an infinite class of unitary transformations between configurationView original: http://arxiv.org/abs/1202.5383
and momentum fractional spaces, thus generalizing the Fourier transform to a
special class of fractal geometries. Each transform diagonalizes a unique
Laplacian operator. We also introduce a new version of fractional spaces, where
coordinates and momenta span the whole real line. In one topological dimension,
these results are extended to more general measures.