Local approximation order to smooth complex valued functions by a finite dimensional space
H
\mathcal {H}
, spanned by certain products of exponentials by polynomials, is investigated. The results obtained, together with a suitable quasi-interpolation scheme, are used for the derivation of the approximation order attained by the linear span of translates of an exponential box spline. The analysis of a typical space
H
\mathcal {H}
is based here on the identification of its dual with a certain space
P
\mathcal {P}
of multivariate polynomials. This point of view allows us to solve a class of multivariate interpolation problems by the polynomials from
P
\mathcal {P}
, with interpolation data characterized by the structure of
H
\mathcal {H}
, and to construct bases of
P
\mathcal {P}
corresponding to the interpolation problem.