Approximation by quasi-projection operators and dual wavelet frames in Sobolev spaces

For a quasi-projection operator $Q_j(f,\phi, \widetilde\phi)$ formed by a compactly supported function $\phi$ and a compactly supported distribution $\widetilde\phi$ with dilation matrix $M$, we establish necessary and sufficient conditions under which it provides prescribed simultaneous approximation and simultaneous density orders for a function $f$ from the Sobolev space $H^s({\mathbb R}^d)$ and for its derivatives. The obtained criteria are then used to endow MRA-based dual wavelet frames in a pair of dual Sobolev spaces with the desired simultaneous approximation properties.