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.

R Radha

Department of Mathematics

Mathematical techniques

Frames

LAURENT OPERATOR

RIESZ BASIS

SHIFT-INVARIANT SPACE

WIENER AMALGAM SPACES

functional analysis

IIT Madras is a public technical and research university located in Chennai, Tamil Nadu. Founded in 1959, it is recognised as an Institute of National Importance.

IIT Madras has been ranked as the top engineering institute in India for four years in a row by the National Institutional Ranking Framework of the MHRD

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IIT Madras

Sampling and Reconstruction in a Shift Invariant Space with Multiple Generators

Numerical Functional Analysis and Optimization

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.

Applicable Analysis

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

Annali di Matematica Pura ed Applicata

Sampling and reconstruction in shift-invariant spaces on Rd

A local reconstruction method is discussed for functions belonging to a shift invariant space arising out of a function which has polynomial decay in time domain and moderate decay in frequency domain such that its Fourier transform is non-vanishing on a unit interval. This method is useful in solving the scaling identity (associated wavelet basis) locally. © 2008 IMACS.

Applied Numerical Mathematics

Local reconstruction of a function from a non-uniform sampled data

Dynamical sampling in multiply generated shift-invariant spaces

Duke Mathematical Journal

Gabor frames and totally positive functions

Proceedings of the American Mathematical Society

Convolution sampling and reconstruction of signals in a reproducing kernel subspace

Journal	Data powered by TypesetNumerical Functional Analysis and Optimization
Publisher	Data powered by TypesetTaylor and Francis Inc.
ISSN	01630563
Open Access	No