Continuous Gabor transform for a class of non-Abelian groups

Qualitative Uncertainty Principle for Gabor Transform

“…Also, using Plancherel theorem [4,Theorem 7.44], we see that f ψ x (π) is a Hilbert-Schmidt operator for almost all π ∈ G. Therefore, G ψ f (x, π) is a Hilbert-Schmidt operator for all x ∈ G and for almost all π ∈ G. As in [3], for f ∈ C c (G) and a window function ψ ∈ L 2 (G), we have…”

“…In [11], it has been proved that for f ∈ L 2 (R) \ {0} and a window function ψ, the support of G ψ f is a set of infinite Lebesgue measure. The continuous Gabor transform for second countable, unimodular and type I group has been defined in [3]. A brief description is given in section 2.…”

Uncertainty principles on nilpotent Lie groups

“…One can verify that G ψ f (x, π) is a Hilbert-Schmidt operator for all x ∈ G and for almost all π ∈ G. We can extend G ψ uniquely to a bounded linear operator from L 2 (G) into a closed subspace of H 2 (G × G) which will be denoted by G ψ . As in [6], for f 1 , f 2 ∈ L 2 (G) and window functions ψ 1 and ψ 2 , we have…”

Hardy’s Theorem for Gabor Transform

“…One can verify that G ψ f (x, π) is a Hilbert-Schmidt operator for all x ∈ G and for almost all π ∈ G. We can extend G ψ uniquely to a bounded linear operator from L 2 (G) into a closed subspace H of H 2 (G × G), which we still denote by G ψ . As in [4], for f 1 , f 2 ∈ L 2 (G) and window functions ψ 1 and ψ 2 ,…”