Left translates of a square integrable function on the Heisenberg group

Abstract

The aim of this paper is to study some properties of left translates of a square integrable function on the Heisenberg group. First, a necessary and sufficient condition for the existence of the canonical dual to a function ϕ ∈ L 2 (R 2n ) is obtained in the case of twisted shift-invariant spaces. Further, characterizations of ℓ 2 -linear independence and the Hilbertian property of the twisted translates of a function ϕ ∈ L 2 (R 2n ) are obtained. Later these results are shown in the case of the Heisenberg group.2010 Mathematics Subject Classification. Primary 42C15; Secondary 43A30, 42B10.

Gabor System Based on the Unitary Dual of the Heisenberg Group

“…In the recent years frames and Riesz bases for system of translates have been studied extensively in various group settings such as locally compact abelian group, compact non-abelian group, Heisenberg group and in general certain Lie groups. (See [15], [4], [21], [7], [18], [13], [20], [1] and [19].) The wavelet system has also been studied for a locally compact abelian group and Heisenberg group.…”

Twisted B-splines in the complex plane

“…This study was later extended to the Heisenberg group in [14]. In [13], Radha and Adhikari studied some properties of twisted translates of a squareintegrable function on C and later extended those results to the Heisenberg group. Recently, in [1], the orthonormality of a wavelet system associated with twisted translates and left translates on the Heisenberg group were investigated.…”