Tikhonov's regularization of the problem of assumption differentiation under uncertainty

In a separable Hilbertian space the problem is stated of checking static assumptions on moment functions of two probabilistic measures of order up to $N$, inclusive, The a priori uncertainty is in small disturbances of the assumptions to be tested. The assumption differentiation problem is shown to be incorrect according to Hadamar. Theorems are given on conditions for existence and stability of solving functionals in the class of Hilbert-Schmidt polynomials. The solving functionals are found by solving a variational problem for the argument for a smoothing Tikhonov filter.