论文标题
功能空间中稳定的相位检索
Stable phase retrieval in function spaces
论文作者
论文摘要
令$（ω，σ，μ）$为度量空间，$ 1 \ leq p \ leq \ infty $。如果存在常数$ c \ geq 1 $，则据说子空间$ e \ subseteq l_p（μ）$可以进行稳定的阶段检索（SPR），因此对于任何$ f，g \ in e $，我们都有$ $ \ inf_ {|λ| = 1} \ | f-λg\ | \ leq c \ || f | - | - | g | \ |。 $$ 在这种情况下，如果已知$ | f | $，则$ f $唯一确定为不可避免的全球相位因子$λ$；此外，相恢复图为$ C $ -LIPSCHITZ。相位检索出现在几种应用的情况下，从晶体学到量子力学。 在本文中，我们构建了各种子空间进行稳定的相位检索，并与$λ（p）$ - 集理论建立连接。此外，我们设定了基础，以分析一般函数空间中稳定相位的检索。特别是，这使我们能够证明Hölder稳定的阶段检索意味着稳定的阶段检索，从而改善了M. Christ和第三和第四作者的近期文章中的稳定界限。我们还表征了那些紧凑的Hausdorff Spaces $ K $，以便$ c（k）$包含无限的尺寸SPR子空间。
Let $(Ω,Σ,μ)$ be a measure space, and $1\leq p\leq \infty$. A subspace $E\subseteq L_p(μ)$ is said to do stable phase retrieval (SPR) if there exists a constant $C\geq 1$ such that for any $f,g\in E$ we have $$ \inf_{|λ|=1} \|f-λg\|\leq C\||f|-|g|\|. $$ In this case, if $|f|$ is known, then $f$ is uniquely determined up to an unavoidable global phase factor $λ$; moreover, the phase recovery map is $C$-Lipschitz. Phase retrieval appears in several applied circumstances, ranging from crystallography to quantum mechanics. In this article, we construct various subspaces doing stable phase retrieval, and make connections with $Λ(p)$-set theory. Moreover, we set the foundations for an analysis of stable phase retrieval in general function spaces. This, in particular, allows us to show that Hölder stable phase retrieval implies stable phase retrieval, improving the stability bounds in a recent article of M. Christ and the third and fourth authors. We also characterize those compact Hausdorff spaces $K$ such that $C(K)$ contains an infinite dimensional SPR subspace.
