论文标题
关于傅立叶相检索的稳定性
On the Stability of Fourier Phase Retrieval
论文作者
论文摘要
阶段检索与从其傅立叶变换$ | \ wideHat {f} | $从其绝对值中恢复函数$ f $有关。我们研究了Lebesgue空间中此问题的稳定性。我们的主要结果表明$$ \ | f-g \ | _ {l^2（\ mathbb {r}^n）} \ leq 2 \ cdot \ | | \ widehat {f} | - | \ widehat {g} | \ | _ {l^2（\ Mathbb {r}^n）} + h_f \ left（\ | f-g-g \ \ |^{} _ {l^p（\ Mathbb {r}^n）} \ right） $ h_f $是明确的非线性功能，具体取决于$ f $的平滑度，$ j $是一个明确的术语，可捕获翻译下的不变性。一个值得注意的方面是，稳定性以$ l^p $为$ 1 \ leq p <2 $的稳定性，而通常，$ l^p $不能用于控制$ l^2 $，稳定性估计具有反向Hölder不平等的风味。似乎可以想象，估计值是最佳的。
Phase retrieval is concerned with recovering a function $f$ from the absolute value of its Fourier transform $|\widehat{f}|$. We study the stability properties of this problem in Lebesgue spaces. Our main results shows that $$ \| f-g\|_{L^2(\mathbb{R}^n)} \leq 2\cdot \| |\widehat{f}| - |\widehat{g}| \|_{L^2(\mathbb{R}^n)} + h_f\left( \|f-g\|^{}_{L^p(\mathbb{R}^n)}\right) + J(\widehat{f}, \widehat{g}),$$ where $1 \leq p < 2$, $h_f$ is an explicit nonlinear function depending on the smoothness of $f$ and $J$ is an explicit term capturing the invariance under translations. A noteworthy aspect is that the stability is phrased in terms of $L^p$ for $1 \leq p < 2$ while, usually, $L^p$ cannot be used to control $L^2$, the stability estimate has the flavor of an inverse Hölder inequality. It seems conceivable that the estimate is optimal up to constants.
