PreprintCC BY-NC-ND 4.0arXiv
Gabor frames with no Gabor duals
Abstract
Let H be an infinite-dimensional Hilbert space and let setxninN be a frame for H. We say that a sequence setyninN is an \emph{alternative dual} of setxninN if x=sumliipxynxn for all x∈ H, with the convergence of the series in the norm of H. We say that a sequence setzninN is a \emph{pseudo-synthesis dual} of setxninN if x=sumliipxxnzn for all x∈ 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 L2(Rd) with infinitely many alternative duals (and pseudo-synthesis duals) such that none of them is a Gabor system
Related
BibTeX
@misc{yu2026gabor,
title = {{Gabor frames with no Gabor duals}},
author = {Pu-Ting Yu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.32246},
eprint = {2609.32246},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.32246},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Gabor frames with no Gabor duals
AU - Pu-Ting Yu
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2609.32246
UR - https://arxiv.org/abs/2609.32246
AB - Let H be an infinite-dimensional Hilbert space and let setx_ninN be a frame for H. We say that a sequence sety_ninN is an \emph{alternative dual} of setx_ninN if x=sumliipxy_nx_n for all x∈ H, with the convergence of the series in the norm of H. We say that a sequence setz_ninN is a \emph{pseudo-synthesis dual} of setx_ninN if x=sumliipxx_nz_n for all x∈ 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
ER -