Shift-invariant system on the Heisenberg Group
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Tusi Mathematical Research Group
ORIGINAL PAPER
Shift-invariant system on the Heisenberg Group S. R. Das1 • R. Radha1 Received: 17 January 2020 / Accepted: 17 October 2020 Ó Tusi Mathematical Research Group (TMRG) 2020
Abstract In this paper, a shift-invariant system of the form fLam gs : m 2 C; s 2 Zg for a [ 0 is studied on the Heisenberg group Hn , where Lx denotes the left translation operator on Hn and C is a lattice in Hn . The characterizations for the shift-invariant system to be a Bessel sequence and a frame sequence are given in terms of certain operators arising from the fiber map corresponding to this system. For a shift invariant system to be a frame sequence (Riesz sequence) the characterization is given in terms of dual Gramian (Gramian). Moreover, the problem of characterizing a pair of shiftinvariant systems to be dual frames is also discussed. Keywords Dual Gramian Frames Gramian Range function Riesz basis Shift-invariant space
Mathematics Subject Classification 42C15 43A30 42B10
1 Introduction A closed subspace V L2 ðRÞ is called a shift-invariant space if f 2 V ) sk f 2 V for any k 2 Z where sx denotes the translation operator sx f ðyÞ ¼ f ðy xÞ. The spaces appearing in the definition of multiresolution analysis in order to construct wavelets are popular examples of shift-invariant spaces. We refer to [14, 15] for further details. If / 2 L2 ðRÞ, then Vð/Þ ¼ spanfsk / : k 2 Zg is usually called the principal shift-invariant space. The shift-invariant spaces have been widely applied Communicated by Gestur Olafsson. & R. Radha [email protected] S. R. Das [email protected] 1
Department of Mathematics, Indian Institute of Technology, Madras, Chennai, India
S. R. Das and R. Radha
in various fields such as Approximation theory, Mathematical Sampling theory, Communication engineering and so on. On the other hand, shift-invariant spaces have also been studied from the theoretical point of view on various group settings. Bownik in [6] obtained characterization of shift-invariant spaces on Rn in terms of range functions. He also obtained characterization for the system of translates to be a frame sequence or a Riesz sequence on Rn . Later these results were studied on locally compact abelian groups in [7, 8, 13] and on a non-abelian compact group in [18]. In [10] Currey et al. characterized shift-invariant spaces in terms of range functions for SI/Z type groups. These are simply connected Lie groups having unitary irreducible representations that are square-integrable modulo the center. Using the range function s, they obtain characterizations for EðAÞ to be frame and Riesz basis in the following sense. They show that EðAÞ is frame iff fsðLk /ÞðrÞ : / 2 A; k 2 C1 g is a frame for JðrÞ for a:e: r 2 Tn . In [4] Barbieri et al. obtained the characterization of frames and Riesz basis consisting of system of left translates using the bracket map on the polarized Heisenberg group. In [16], Radha and Adhikari studied these types of
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