On the Truncation Error of Generalized Sampling Expansions in Shift-Invariant Spaces

2007 ◽  
Vol 6 (1) ◽  
pp. 53-69
Author(s):  
Antonio G. García ◽  
G. Pérez-Villalón
Keyword(s):  
Truncation Error ◽  
Generalized Sampling ◽  
Shift Invariant Spaces ◽  
Invariant Spaces ◽  
Sampling Expansions

MULTI-CHANNEL SAMPLING ON SHIFT-INVARIANT SPACES WITH FRAME GENERATORS

2012 ◽  
Vol 10 (01) ◽  
pp. 1250003 ◽  
Author(s):  
A. G. GARCIA ◽  
J. M. KIM ◽  
K. H. KWON ◽  
G. J. YOON
Keyword(s):  
Continuous Function ◽  
Sufficient Conditions ◽  
Sampling Theory ◽  
Invariant Space ◽  
Frame Sequence ◽  
Shift Invariant Spaces ◽  
Necessary And Sufficient ◽  
Invariant Spaces ◽  
Sampling Expansions ◽  
Shift Invariant Space

Let φ be a continuous function in L2(ℝ) such that the sequence {φ(t - n)}n∈ℤ is a frame sequence in L2(ℝ) and assume that the shift-invariant space V(φ) generated by φ has a multi-banded spectrum σ(V). The main aim in this paper is to derive a multi-channel sampling theory for the shift-invariant space V(φ). By using a type of Fourier duality between the spaces V(φ) and L2[0, 2π] we find necessary and sufficient conditions allowing us to obtain stable multi-channel sampling expansions in V(φ).

SAMPLING EXPANSION IN SHIFT INVARIANT SPACES

2008 ◽  
Vol 06 (02) ◽  
pp. 223-248 ◽  
Author(s):  
J. M. KIM ◽  
K. H. KWON
Keyword(s):  
Truncation Error ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Sufficient Conditions ◽  
Sufficient Condition ◽  
Expansion Formula ◽  
Sampling Series ◽  
Shift Invariant Spaces ◽  
Necessary And Sufficient ◽  
Invariant Spaces

For any ϕ(t) in L2(ℝ), let V(ϕ) be the closed shift invariant subspace of L2(ℝ) spanned by integer translates {ϕ(t - n) : n ∈ ℤ} of ϕ(t). Assuming that ϕ(t) is a frame or a Riesz generator of V(ϕ), we first find conditions under which V(ϕ) becomes a reproducing kernel Hilbert space. We then find necessary and sufficient conditions under which an irregular or a regular shifted sampling expansion formula holds on V(ϕ) and obtain truncation error estimates of the sampling series. We also find a sufficient condition for a function in L2(ℝ) that belongs to a sampling subspace of L2(ℝ). Several illustrating examples are also provided.

scholarly journals Generalized sampling in shift-invariant spaces with multiple stable generators

2008 ◽  
Vol 337 (1) ◽  
pp. 69-84 ◽  
Author(s):  
A.G. García ◽  
M.A. Hernández-Medina ◽  
G. Pérez-Villalón
Keyword(s):  
Generalized Sampling ◽  
Shift Invariant Spaces ◽  
Invariant Spaces
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