caa a22 4500 605496099 CHVBK 20210128100534.0 cr unu---uuuuu 210128e20151201xx s 000 0 eng 10.1007/s10231-014-0439-x doi (NATIONALLICENCE)springer-10.1007/s10231-014-0439-x Sampling and reconstruction in shift-invariant spaces on $$\mathbb {R}^d$$ R d [Elektronische Daten] [A. Selvan, R. Radha] Let $$\phi \in W(C,\ell ^1)$$ ϕ ∈ W ( C , ℓ 1 ) such that $$\{\tau _n\phi :n\in \mathbb {Z}^d\}$$ { τ n ϕ : n ∈ Z d } forms a Riesz basis for $$V(\phi )$$ V ( ϕ ) . It is shown that $$\mathbb {Z}^d$$ Z d is a stable set of sampling for $$V(\phi )$$ V ( ϕ ) if and only if $$\Phi ^\dagger (x)\ne 0$$ Φ † ( x ) ≠ 0 , for every $$x\in \mathbb {T}^d$$ x ∈ T d , where $$\Phi ^{\dagger }(x):=\sum _{n\in \mathbb {Z}^d}\phi (n)e^{2\pi in\cdot x},~~ x\in \mathbb {T}^d$$ Φ † ( x ) : = ∑ n ∈ Z d ϕ ( n ) e 2 π i n · x , x ∈ T d . Sampling formulae are provided for reconstructing a function $$f\in V(\phi )$$ 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. Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin Heidelberg, 2014 Frames nationallicence Laurent operator nationallicence Riesz basis nationallicence Shift-invariant space nationallicence Wiener amalgam space nationallicence Zak transform nationallicence Selvan A. Department of Mathematics, Indian Institute of Technology Madras, 600 036, Chennai, India aut Radha R. Department of Mathematics, Indian Institute of Technology Madras, 600 036, Chennai, India aut Annali di Matematica Pura ed Applicata (1923 -) Springer Berlin Heidelberg 194/6(2015-12-01), 1683-1706 0373-3114 194:6<1683 2015 194 10231 https://doi.org/10.1007/s10231-014-0439-x text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s10231-014-0439-x text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Selvan A. Department of Mathematics, Indian Institute of Technology Madras, 600 036, Chennai, India aut NATIONALLICENCE 700 1- Radha R. Department of Mathematics, Indian Institute of Technology Madras, 600 036, Chennai, India aut NATIONALLICENCE 773 0- Annali di Matematica Pura ed Applicata (1923 -) Springer Berlin Heidelberg 194/6(2015-12-01), 1683-1706 0373-3114 194:6<1683 2015 194 10231