  Several properties of stationary subdivision schemes are nowadays well
understood. In particular, it is known that the polynomial generation and
reproduction capability of a stationary subdivision scheme is strongly
connected with sum rules, its convergence, smoothness and approximation order.
The aim of this paper is to show that, in the non-stationary case, exponential
polynomials and approximate sum rules play an analogous role of polynomials and
sum rules in the stationary case. Indeed, in the non-stationary univariate case
we are able to show the following important facts: i) reproduction of $N$
exponential polynomials implies approximate sum rules of order $N$; ii)
generation of $N$ exponential polynomials implies approximate sum rules of
order $N$, under the additional assumption of asymptotical similarity and
reproduction of one exponential polynomial; iii) reproduction of an
$N$-dimensional space of exponential polynomials and asymptotical similarity
imply approximation order $N$; iv) the sequence of basic limit functions of a
non-stationary scheme reproducing one exponential polynomial converges
uniformly to the basic limit function of the asymptotically similar stationary
scheme.
