Relative Openness of the Gabor Frame Set on the Hyperbolas

作者：A. Kulikov · 发表于：Journal of Fourier Analysis and Applications · 年份：2025 · DOI：10.1007/s00041-024-10141-8 · 被引用次数：1

We prove that for the functions of the form g(x)=h(x)+Cx+i\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g(x) = h(x) + \frac{C}{x+i}$$\end{document}, where h belongs to the continuous Wiener algebra W0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W_0$$\end{document}, the intersection of the Gabor frame set Fg\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {F}_g$$\end{document} with every hyperbola {α,β>0∣αβ=c}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{\alpha , \beta > 0 \mid \alpha \beta = c\}$$\end{document} is open in the relative topology. In particular, this applies to all rational functions g. We also show that the same phenomenon occurs for the functions from the Wiener algebra W which are discontinuous at one point.