"A Classification of Lattices for Multivariate Gabor Analysis via Symplectic Forms"Gjertsen, MichaelThe theory of multivariate Gabor frames is much more unwieldy than the univariate theory, in large part due to the multitude of possible lattices on the time-frequency plane $\mathbb R^{2d}$ when $d \geq 2$. We introduce an equivalence relation on the set of all lattices in $\mathbb R^{2d}$ such that equivalent lattices share identical structures of Gabor frames (up to unitary equivalence—a notion we define). These equivalence classes are parameterized by invertible skew-symmetric $2d \times 2d$ matrices over $\mathbb R$, which means that $2d^2 - d$ parameters suffice to describe the possible behaviors of Gabor frames on $\mathbb R^d$—this is less than half of the $(2d)^2$ degrees of freedom in the choice of lattice. The invertible skew-symmetric matrices are precisely those matrices that represent symplectic forms on the time-frequency plane, and indeed, the theory of symplectic and metaplectic transformations plays a crucial role. In fact, the behavior of Gabor systems is invariant under symplectic transformations of the time-frequency plane, and the converse is true as well:\ linear transformations of the time-frequency plane which preserve the structure of Gabor systems are necessarily symplectic. This identifies symplectic transformations as the structure-preserving transformations for Gabor theory and thus deepens the link between Gabor theory and symplectic geometry. |
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