For the noncommutative 2-torus A_α, we have recently shown (FF and L.Suriano) that it is possible to construct explicitly type III representations π, at least when α is Liouville number. In addition, if α is Liouville, with a faster approximation property, it is possible to construct also genuine (i.e. non trivial) modular spectral triples.

For such examples, we introduce and investigate a one-parameter family with parameter t∈
$$extract_itex$$0,1$$/extract_itex$$ , of Fourier transforms. Also, we prove the analogous of Riemann-Lebesgue Lemma and Hausdorff-Young Theorem. Finally, for p∈
$$extract_itex$$1,2$$/extract_itex$$ we establish an inversion formula arising from the Cesaro mean and the Poisson average.