On density order of quasi-projection operators and dual wavelet frames

From two functions in the Hilbert space \(L^2(\mathbb {R}^d)\) $L^2({\mathbb{R}}^d)$ whose Fourier transform has certain decay, one defines a quasi-projection operator. In this paper, we prove necessary and sufficient conditions on those functions in order to the associated quasi-projection operator provides a desired approximation order and density order. To give our conditions we will use the classical notion of approximate continuity. As a consequence, we obtain approximation properties of dual wavelet frame constructed by Mixed Oblique Extension Principle. We show our results in the context of reducing subspaces of \(L^2(\mathbb {R}^d)\) $L^2({\mathbb{R}}^d)$ with a dilation given by an expansive linear map preserving the integer lattice.