AI 中文总结
研究Sobolev空间中拟投影算子为函数及其导数提供同时逼近和密度阶的条件，利用所得准则赋予对偶Sobolev空间中基于MRA的对偶小波框架所需同时逼近性质。
AI 中文摘要
对于由紧支集函数$\phi$和具有伸缩矩阵$M$的紧支集分布$\widetilde\phi$构成的拟投影算子$Q_j(f,\phi,\widetilde\phi)$，我们建立了充分必要条件，使其能为Sobolev空间$H^s(\mathbb{R}^d)$中的函数$f$及其导数提供规定的同时逼近和同时密度阶。所得准则随后用于赋予一对对偶Sobolev空间中基于MRA的对偶小波框架所需的同时逼近性质。
英文摘要
For a quasi-projection operator $Q_j(f,ϕ, \widetildeϕ)$ formed by a compactly supported function $ϕ$ and a compactly supported distribution $\widetildeϕ$ 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.
