Year: 2026
Type: preprint
Abstract: 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
Source: arXiv (Cornell University)
Author: Yu, Pu-Ting
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214752762