For a compactly supported function
φ
\varphi
in
R
d
{\mathbb {R}^d}
we study quasiinterpolants based on point evaluations at the integer lattice. We restrict ourselves to the case where the coefficient sequence
λ
f
\lambda f
, for given data f, is computed by applying a univariate polynomial q to the sequence
φ
|
Z
d
\varphi {|_{{\mathbb {Z}^d}}}
, and then convolving with the data
f
|
Z
d
f{|_{{\mathbb {Z}^d}}}
. Such operators appear in the well-known Neumann series formulation of quasi-interpolation. A criterion for the polynomial q is given such that the corresponding operator defines a quasi-interpolant. Since our main application is cardinal interpolation, which is well defined if the symbol of
φ
\varphi
does not vanish, we choose q as the partial sum of a certain Faber series. This series can be computed recursively. By this approach, we avoid the restriction that the range of the symbol of
φ
\varphi
must be contained in a disk of the complex plane excluding the origin, which is necessary for convergence of the Neumann series. Furthermore, for symmetric
φ
\varphi
, we prove that the rate of convergence to the cardinal interpolant is superior to the one obtainable from the Neumann series.