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

Abstract

For a quasi-projection operator $Q_j(f,\ddot{I}•,\tilde{\ddot{I}}•)$ formed by a compactly supported function $\ddot{I}•$ and a compactly supported distribution $\tilde{\ddot{I}}•$ 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.