On a class of non-uniform average sampling expansions and partial reconstruction in subspaces of L 2(ℝ)

Abstract

Let φ be a function in the Wiener amalgam space W ∞ (L 1 ) with a non-vanishing property in a neighborhood of the origin for its Fourier transform φ, τ = {τ n } n∈Z be a sampling set on R and V τ φ be a closed subspace of L 2 (R) containing all linear combinations of τ -translates of φ. In this paper we prove that every function f ∈ V τ φ is uniquely determined by and stably reconstructed from the sample set L τAs our reconstruction formula involves evaluating the inverse of an infinite matrix we consider a partial reconstruction formula suitable for numerical implementation. Under an additional assumption on the decay rate of φ we provide an estimate to the corresponding error.

Sampling and Approximation in Shift Invariant Subspaces of $$L_2(\mathbb {R})$$

“…For well-posedness, we need to determine a certain subspace of L 2 where f lives in and require a space for the sample vector c f to stay. For more about average sampling expansions, we refer to [1,3,4,7,9,30,36,37] and related references therein.…”

“…Proof A more detailed proof is demonstrated in [7]. For any f ∈ V φ , and for any x ∈ X , where X is a bounded interval of R, we have…”

Random sampling and reconstruction of concentrated signals in a reproducing kernel space

“…generated by (non)uniform shifts of the Gaussian function exp(−x 2 ), where Φ := {φ i (x) = exp(−(x − i − θ i ) 2 ), i ∈ Z} and θ i ∈ [−1/10, 1/10], i ∈ Z, are randomly selected [10,20,35,47]. Our numerical simulations indicate that the correlation matrix A Φ := ( φ i , φ j ) i,j∈Z has bounded inverse on ℓ 2 , and hence the inverse A −1 Φ = (b ij ) i,j∈Z has polynomial off-diagonal decay of any order by Wiener's lemma for infinite matrices [29,32,43,46,48].…”

On the structure and interpolation properties of quasi shift-invariant spaces

“…Let I ψ be the interpolation operator associated with the generator ψ.Then if f ∈ V (ψ), f = I ψ f .Proof. Note that by(5) and Theorem 2(i),I ψ f ∈ V (ψ). Moreover, I ψ f is the unique function in V (ψ) such that I ψ f (k) = f (k).However, evidently f ∈ V (ψ) satisfies this relation as well; consequently I ψ f = f .…”