Assume that \(\varphi \in L^2({\mathbb R})\) fulfils a natural decay condition and the Strang-Fix condition and that the system \(\{\varphi(x-k);\, k \in {\mathbb Z}\}\) is orthonormal. Note that \(\varphi\) is not necessarily a scaling function. The authors prove polynomial reproducing properties with absolute and uniform convergence on arbitrary compact subsets of \({\mathbb R}\). This approach leads to a new exact formula for the computation of moments of even order.