$α$ -剪切波的分析与合成稀疏性
Analysis vs. synthesis sparsity for $α$-shearlets
摘要：
有两个与帧$Ψ=(ψ_i)_{i\in I}$相关的稀疏性概念：$f$的分析稀疏性意味着分析系数$(\langle f,ψ_i\rangle)_i$是稀疏的，而合成稀疏性意味着$f=\sum_i c_iψ_i$具有稀疏系数$(c_i)_i$。这里，$c=(c_i)_i$的稀疏性意味着给定$p<2$的$c\in\ell^p(I)$。我们表明，这两个概念的稀疏性相吻合，如果$Ψ={\rm SH}(φ,ψ;δ)$是一个离散的（锥适应）剪切帧与“好”发电机$φ,ψ$和足够精细的采样密度$δ>0$。所需的“尼斯”是明确量化的傅立叶衰减和消失的时刻条件。准确地说，我们发现，合适的剪切波系统同时提供Banach框架和原子分解的剪切波光滑空间$\mathscr{S}_s^{p,q}$引入Labate等人。因此，成员在$\mathscr{S}_s^{p,q}$是同时相当于分析稀疏和合成稀疏w.r.t.剪羊毛的框架。作为一个应用程序，我们证明了剪切产生（几乎）最佳逼近率的卡通函数$f$：如果$ε>0$，那么$\Vert f-f_N\Vert_{L^2}\lesssim N^{-(1-ε)}$，其中$f_N$是一个线性组合的N剪切。这似乎是众所周知的，但现有的证明只建立这个近似率w.r.t. $Ψ$的双$\tildeΨ$，而不是w.r.t. $Ψ$本身。这并不完全令人满意，因为$\tildeΨ$的属性（衰减、平滑度等）大部分都是未知的。我们也考虑$α$剪切系统。对于这些，剪切波平滑空间必须由$α$剪切波平滑空间代替。我们完全刻画这些空间之间的嵌入，使我们能够决定是否稀疏w.r.t. $α_1$ -剪切波意味着稀疏性w.r.t. $α_2$ -羊毛衫。
英文摘要：
There are two notions of sparsity associated to a frame $Ψ=(ψ_i)_{i\in I}$: Analysis sparsity of $f$ means that the analysis coefficients $(\langle f,ψ_i\rangle)_i$ are sparse, while synthesis sparsity means that $f=\sum_i c_iψ_i$ with sparse coefficients $(c_i)_i$. Here, sparsity of $c=(c_i)_i$ means $c\in\ell^p(I)$ for a given $p<2$. We show that both notions of sparsity coincide if $Ψ={\rm SH}(φ,ψ;δ)$ is a discrete (cone-adapted) shearlet frame with 'nice' generators $φ,ψ$ and fine enough sampling density $δ>0$. The required 'niceness' is explicitly quantified in terms of Fourier-decay and vanishing moment conditions. Precisely, we show that suitable shearlet systems simultaneously provide Banach frames and atomic decompositions for the shearlet smoothness spaces $\mathscr{S}_s^{p,q}$ introduced by Labate et al. Hence, membership in $\mathscr{S}_s^{p,q}$ is simultaneously equivalent to analysis sparsity and to synthesis sparsity w.r.t. the shearlet frame. As an application, we prove that shearlets yield (almost) optimal approximation rates for cartoon-like functions $f$: If $ε>0$, then $\Vert f-f_N\Vert_{L^2}\lesssim N^{-(1-ε)}$, where $f_N$ is a linear combination of N shearlets. This might appear to be well-known, but the existing proofs only establish this approximation rate w.r.t. the dual $\tildeΨ$ of $Ψ$, not w.r.t. $Ψ$ itself. This is not completely satisfying, since the properties of $\tildeΨ$ (decay, smoothness, etc.) are largely unknown. We also consider $α$-shearlet systems. For these, the shearlet smoothness spaces have to be replaced by $α$-shearlet smoothness spaces. We completely characterize the embeddings between these spaces, allowing us to decide whether sparsity w.r.t. $α_1$-shearlets implies sparsity w.r.t. $α_2$-shearlets.
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