无Gabor对偶的Gabor框架
Gabor frames with no Gabor duals
AI总结：
本文证明在任意光滑度下，存在$L^2(\R^d)$上的Gabor框架，其拥有无限多个替代对偶和伪合成对偶，但均非Gabor系统。
AI中文摘要：
设$H$为无限维Hilbert空间，且设$\set{x_n}\inN$为$H$的一个框架。若对任意$x\in H$，有$$x=\sumli\ip{x}{y_n}x_n$$且级数在$H$的范数下收敛，则称序列$\set{y_n}\inN$为$\set{x_n}\inN$的\emph{替代对偶}。若对任意$x\in H$，有$$x=\sumli\ip{x}{x_n}z_n$$且级数在$H$的范数下收敛，则称序列$\set{z_n}\inN$为$\set{x_n}\inN$的\emph{伪合成对偶}。我们证明，在生成元的任意光滑度水平下，存在$L^2(\R^d)$上的Gabor框架，其具有无限多个替代对偶（及伪合成对偶），且其中任何一个都不是Gabor系统。
英文摘要：
Let $H$ be an infinite-dimensional Hilbert space and let $\set{x_n}\inN$ be a frame for $H$. We say that a sequence $\set{y_n}\inN$ is an \emph{alternative dual} of $\set{x_n}\inN$ if $$x=\sumli\ip{x}{y_n}x_n\quad \text{for all }x\in H,$$ with the convergence of the series in the norm of $H.$ We say that a sequence $\set{z_n}\inN$ is a \emph{pseudo-synthesis dual} of $\set{x_n}\inN$ if $$x=\sumli\ip{x}{x_n}z_n \quad \text{for all }x\in H,$$ with the convergence of the series in the norm of $H.$ We prove that, at every level of smoothness of the generator, there exist Gabor frames for $L^2(\R^d)$ with infinitely many alternative duals (and pseudo-synthesis duals) such that none of them is a Gabor system
