  To each arbitrary given general geometric structure on $\mathbb{R}^{n}$, we
associate a pair of compatible Fourier transforms, that prove to appear
naturally in the framework of Poisson's summation formula for full lattices. We
study their properties and the compatibility with the classical $n-$dimensional
Fourier transform. In the case of a positive definite geometric structure, we
show that these geometric Fourier transforms induce a geometric fractional
Laplacian, with properties similar to those of the classical fractional
Laplacian.
