Abstract
The modulation spaces (Rd) quantify the time-frequency concentration of functions and distributions. The first main result of this paper proves embeddings of certain modulation spaces into VMO, the space of functions with vanishing mean oscillation. The second main result proves that the Zak transform maps certain modulation spaces on Rd into modulation spaces on R2d. These two results allow us to give a Balian–Low-type of uncertainty principle for Gabor systems in the setting of modulation spaces.
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