A Note on the Invertibility of the Gabor Frame Operator on Certain Modulation Spaces

作者：Dae Gwan Lee, F. Philipp, F. Voigtlaender · 发表于：Journal of Fourier Analysis and Applications · 年份：2021 · DOI：10.1007/s00041-022-09980-0 · 被引用次数：1 · 研究领域：Mathematics

We consider Gabor frames generated by a general lattice and a window function that belongs to one of the following spaces: the Sobolev space V1=H1(Rd)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V_1 = H^1(\mathbb {R}^d)$$\end{document}, the weighted L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document}-space V2=L1+|x|2(Rd)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V_2 = L_{1 + |x|}^2(\mathbb {R}^d)$$\end{document}, and the space V3=H1(Rd)=V1∩V2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V_3 = \mathbb {H}^1(\mathbb {R}^d) = V_1 \cap V_2$$\end{document} consisting of all functions with finite uncertainty product; all these spaces can be described as modulation spaces with respect to suitable weighted L2\documentclass[12pt]{minimal} \use...