Gabor systems with Hermite functions of order N and oversampling greater than N+1 which are not frames

Key Points

• The results show that certain density conditions do not guarantee the frame property in Gabor systems.
• A key finding illustrates how the Zak transform's zeros influence the frame property of oversampled systems.
• This research addresses limitations of existing conditions, offering new insights into Gabor systems.
• The implications suggest that current criteria for frame properties should be re-evaluated for oversampled Gabor systems.