In this paper, it is shown that for a fixed m ∈ N, Z/m is a stable set of sampling for V(φ 1 , φ 2 , . . . , φ s ) if and only if there exists γ &gt; 0 such that | detΦ(x)| ≥ γ a.e. on T, where Φ is a symbol matrix of a block Laurent operator associated with the sample set of φ l , l = 1, . . . , s. A similar result is proved when rZ where r = p/q, p, q ∈ N and gcd(p, q) = 1,is a stable set of sampling for V(φ) (single generator) as well as V(φ 1 , φ 2 , . . . , φ s ) (multi generators). A perturbation of stable set of sampling is also discussed. © 2019, © 2019 Informa UK Limited, trading as Taylor & Francis Group.

R Radha

Department of Mathematics

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IIT Madras

Invertibility of Laurent operators and shift invariant spaces with finitely many generators

Applicable Analysis

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.

Fulltext

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

The aim of this paper is to study sampling and reconstruction in a shift invariant space with multiple generators. On contrary to the classical case of a shift invariant space with a single generator, it is shown that Z cannot be a stable set of sampling for V(Φ) where Φ=Φ{ 1 , 2 ,..., n } when r ≥ 2. Further the problems of perturbation of a stable set of sampling and local reconstruction method are discussed along with an illustration and implementation. © 2018, © 2018 Taylor & Francis Group, LLC.

Numerical Functional Analysis and Optimization

Sampling and Reconstruction in a Shift Invariant Space with Multiple Generators

Let ϕ∈W(C,ℓ1) such that {τnϕ:n∈Zd} forms a Riesz basis for V(ϕ). It is shown that Zd is a stable set of sampling for V(ϕ) if and only if Φ†(x)≠0, for every x∈Td, where Φ†(x):=∑n∈Zdϕ(n)e2πin·x,x∈Td. Sampling formulae are provided for reconstructing a function f∈V(ϕ) from uniform samples using Zak transform and complex analytic technique. The problem of sampling and reconstruction is discussed in the case of irregular samples also. The theory is illustrated with some examples, and numerical implementation for reconstruction of a function from its nonuniform samples is provided using MATLAB. © 2014, Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin Heidelberg.

Journal	Data powered by TypesetApplicable Analysis
Publisher	Data powered by TypesetTaylor and Francis Ltd.
ISSN	00036811
Open Access	No