  The core of quantum tomography is the possibility of writing a generally
unbounded complex operator in form of an expansion over operators that are
generally nonlinear functions of a generally continuous set of spectral
densities--the so-called "quorum" of observables. The expansion is generally
non unique, the non unicity allowing further optimization for given criteria.
The mathematical problem of tomography is thus the classification of all such
operator expansions for given (suitably closed) linear spaces of unbounded
operators--e.g. Banach spaces of operators with an appropriate norm. Such
problem is a difficult one, and remains still open, involving the theory of
general basis in Banach spaces, a still unfinished chapter of analysis. In this
paper we present new nontrivial operator expansions for the quorum of
quadratures of the harmonic oscillator, and introduce a first very preliminary
general framework to generate new expansions based on the Kolmogorov
construction. The material presented in this paper is intended to be helpful
for the solution of the general problem of quantum tomography in infinite
dimensions, which corresponds to provide a coherent mathematical framework for
operator expansions over functions of a continuous set of spectral densities.
