Quantitative versions of Whitney problems require construction of functions with prescribed values on a given finite subset E ⊂ Rn, which minimize a preferred functional norm. After solving this minimization problem we want to compute some natural functionals of the minimizer, e.g., its values at other points. Quantization is an art of replacing a minimization problem by a problem of computing certain amplitudes (similar to expected values) for a system where the preferred functional norm is treated as an action functional. There is an interesting connection between the computation of amplitudes (which are represented as functional integrals) and computations of convolutions of functions on some important unipotent Lie groups, similar to the Heisenberg-Weyl groups.