We define an infinite class of unitary transformations between position 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.
"Momentum transforms and Laplacians in fractional spaces." Adv. Theor. Math. Phys. 16 (4) 1315 - 1348, August 2012.