Irregular wavelet/Gabor frames
Regular Gabor frames for \(L^2(\mathbb{R})\) have the form \(\{e^{imbx}g(x-na)\}_{m,n\in \mathbb{Z}}\) for suitable parameters \(a,b>0\) and \(g\in L^2(\mathbb{R})\); they are relatively well understood by now. Using perturbation methods, the authors obtain conditions implying that an irregular Gabor system \(\{e^{i \lambda_{m,n}x}g(x-\mu_{m,n})\}_{m,n\in \mathbb{Z}}\) is a frame. In particular, certain decay and regularity conditions on \(g\) leads to the frame property for the irregular system if \(\{e^{imbx}g(x-na)\}_{m,n\in \mathbb{Z}}\) is a frame for some \(a,b>0\) for which \(\sup |\lambda_{m,n}-mb|\) and \(\sup |\mu_{m,n}-na|\) are sufficiently small. Similar results are derived for irregular wavelet frames.
scientific article; zbMATH DE number 1891391
scientific article