The hypothesis of simultaneous conformal compactification of both space-time and momentum space, possibly identified as two homogeneous spaces of the conformal group, leads to the need to define on them two dual finite lattices correlated by conformal inversions. It is shown that, with the help of orthonormal sets of (Hahn) polynomials onSn identifying with spherical harmonics in the limit of dense lattices, they may be effectively constructed. In fact, they build up examples of orthonormal basis allowing the formulation of both discrete Fourier transforms on the finite, dual lattices and of scale-invariant propagators, which, in the Euclidean case, are shown to be free from both infrared and ultraviolet divergences. In the one-dimensional case, an exactly soluble toy model is presented where the truncation of infinite sums is correlated with the origin of dual, finite lattices. Such lattices carry an action of quantum deformation of the conformal algebraSU(1,1) withq being a root of unity.