In this paper, twisted shift-invariant spaces in L2(ℝ2n) are introduced. Characterizations of orthonormal system and Bessel sequence of twisted translates in L2(ℝ2n) are obtained in terms of Weyl transform. These results appear to be totally different from the classical case of characterizations of orthonormal systems and Bessel sequences of translates in L2(ℝn). Further, characterizations of frames, Riesz basis in twisted shift-invariant spaces in L2(ℝ2n)are also obtained. © 2015 Elsevier Inc.

R Radha

Department of Mathematics

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IIT Madras

Journal of Mathematical Analysis and Applications

On the real line, one of the interesting problems in wavelet analysis is to obtain a characterization for the orthonormality of a wavelet system. In this paper, we wish to characterize the orthonormality of the wavelet system on the Heisenberg group. We also attempt to study the completeness of this system. Further, we consider the wavelet system associated with twisted translations and dilations on L2(C) and study its orthonormality. © 2019 Elsevier Masson SAS

Journal des Mathematiques Pures et Appliquees

Orthonormality of wavelet system on the Heisenberg group

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∈ L2(R2n) is obtained in the case of twisted shift-invariant spaces. Further, characterizations of ℓ2-linear independence and the Hilbertian property of the twisted translates of a function φ∈ L2(R2n) are obtained. Later these results are shown in the case of the Heisenberg group. © 2019, Universitat de Barcelona.

Collectanea Mathematica

Left translates of a square integrable function on the Heisenberg group

Let G be a second countable locally compact abelian group. The aim of this paper is to characterize the left translates on the Heisenberg group [Formula presented] to be frames and Riesz bases in terms of the group Fourier transform. © 2018 Royal Dutch Mathematical Society (KWG)

Indagationes Mathematicae

Frames and Riesz bases for shift invariant spaces on the abstract Heisenberg group

Recently, a characterization of frames in twisted shift-invariant spaces in L2(R2n) has been obtained in [16]. Using this result,we prove a sampling theorem on a subspace of a twisted shift-invariant space in this paper. © 2017 Walter de Gruyter GmbH, Berlin/Boston.

Advances in Pure and Applied Mathematics

A sampling theorem for the twisted shift-invariant space

We study the theory of shift invariant spaces in L2(G), where G is a compact group. We define a range function and show that a relation between an H-invariant space and the range function is valid as in the case of abelian group setting. Here H is assumed to be a closed normal subgroup of G. We also obtain a decomposition for an H-invariant space in terms of principle H-invariant spaces whose generators give rise to "generalized Parseval frames" and use this result to study H-preserving operators. © 2012 Elsevier Masson SAS.

Journal	Journal of Mathematical Analysis and Applications
Publisher	Academic Press Inc.
ISSN	0022247X
Open Access	Yes