On distance in total variation between image measures

We are interested in the estimation of the distance in total variation Δ≔‖Pf(X)−Pg(X)‖varbetween distributions of random variables f(X) and g(X) in terms of proximity of f and g. We propose a simple general method of estimating Δ. For Gaussian and trigonometrical polynomials it gives an asymptotically optimal result (when the degree tends to ∞).