Frames and Riesz bases of twisted shift-invariant spaces in L2(R2n)

Left translates of a square integrable function on the Heisenberg group

“…In [13], Luef provided a connection between the construction of projections in non-commutative tori and the construction of tight Gabor frames for L 2 (R). Recently, the authors obtained characterizations of orthonormal system, Bessel sequence, frame and Riesz basis of twisted shift-invariant spaces in terms of the kernel of the Weyl transform in [10]. Similar characterizations are obtained in the shiftinvariant spaces associated with a countably many mutually orthogonal generators on the Heisenberg group in [11].…”

“…For a study of of frames we refer to [5] and [7]. We shall make use of the following definitions and results which were given in [10].…”

Orthonormality of wavelet system on the Heisenberg group

“…In [10], Radha and Adhikari proved the following We shall now state a similar result in the context of twisted wavelet system. Theorem 3.3.…”

“…Characterization of the orthonormality of a system of translates on the polarised Heisenberg group was studied in [1] using the concept of bracket map. The shift invariant spaces associated with the twisted translations for L 2 (C n ) were recently studied by Radha and Adhikari in [10]. In [11], characterizations of Bessel sequences, orthonormal bases, frames and Riesz bases were studied on the Heisenberg group for a shift invariant space with countably many mutually orthogonal generators.…”

Shift-invariant Spaces with Countably Many Mutually Orthogonal Generators on the Heisenberg group

“…The proofs of these results are obtained by using such type of results which were proved for twisted shift-invariant spaces and characterized in terms of the kernel of the Weyl transform. We refer to [10] of the authors in this connection. As in [3], we introduce a map T from L 2 (H n ) into L 2 ((0, 1], ℓ 2 (Z, B 2 )), using the group Fourier transform on H n , which turns out to be an isometric isomorphism.…”