Asymptotic pointwise error estimates for reconstructing shift-invariant signals with generators in a hybrid-norm space

Sampling and reconstruction of signals in a shift-invariant space are generally studied under the requirement that the generator is in a stronger Wiener amalgam space, and the error estimates are usually given in the sense of Lp,1/ω-norm. Since we often need to reflect the local characteristics of reconstructing error, the asymptotic pointwise error estimates for nonuniform and average sampling in a non-decaying shift-invariant space are discussed under the assumption that the generator is in a hybrid-norm space. Based on Lemma 2.1–Lemma 2.6, we first rewrite the iterative reconstruction algorithms for two kinds of average sampling functionals and prove their convergence. Then, the asymptotic pointwise error estimates are presented for two algorithms under the case that the average samples are corrupted by noise.


Introduction
The classical Shannon sampling theorem shows that a bandlimited signal which lives in the shift-invariant space generated by the sinc function can be recovered from its samples {f (nδ)} n∈Z when the gap δ is small enough [1]. Since the sinc function has infinite support and slow decay, the space of bandlimited functions is often unsuitable for numerical implementations. Retaining some of the simplicity and structure of bandlimited models, sampling in non-bandlimited shift-invariant spaces is more amenable and realistic for many applications [2][3][4][5][6][7][8][9][10][11][12][13]. Sampling and reconstruction of signals in a shift-invariant is generally studied under the condition that the generator ϕ is in a Winner amalgam space W 1,∞,ω (R d ), which is defined as Throughout the paper, the weighting function denoted by ω is always assumed to be continuous, symmetric, positive, and submultiplicative, (1. 2) The decaying weight 1/ω controls the growing rate of the signals living in V p,1/ω (ϕ).
Recently, regular and ideal sampling [7], nonuniform and average sampling [11] have been restudied under a weaker condition that ϕ is in the weighted hybrid-norm space W 1,q,ω (R d ) with q = max{p, p }, p is the conjugate number of p. Here, for 1 ≤ p, q < ∞, the weighted hybrid-norm space is defined as If p or q is infinity, usual adjustments are used. Both [11] and [7] gave the error estimates in the sense of L p,1/ω -norm, but such estimation can only reflect the mean error information.
In some cases, we need to know the local error information. In this paper, we mainly study the asymptotic pointwise error estimates for signals in V p,1/ω (ϕ) under the assumptions (i) ϕ ∈ W 1,∞,ω (R d ).
(ii) lim δ→0 ω δ (ϕ) W 1,∞,ω = 0, where ω δ (ϕ) is the continuous modulus defined by (iii) There exist positive constants A and B such that In fact, it is easy to verify that assumptions (i)-(ii) are satisfied if ϕ ∈ W 1,∞,ω (R d ) is a continuous function, cf. [4]. The sampling set ⊂ R d is assumed to be relatively-separated, that is, for some δ > 0. Furthermore, δ > 0 is said to be a gap of a relatively-separated subset if Given a relatively-separated sampling set , two kinds of average sampling schemes are considered. The first one is (1.4) where the average sampling functionals {ψ γ : γ ∈ } satisfy the following: for some a > 0. Note that the first sampling scheme requires that the sampling functions have compact support, we consider the second average sampling scheme which is defined as , and ψ * a (·) = ψ a (-·). This paper is organized as follows. In Sect. 2, some necessary lemmas are provided for the subsequent sections. In Sect. 3, the iterative reconstruction algorithms are rewritten and their convergence is proved. The asymptotic pointwise error estimates for both average sampling schemes are presented in Sect. 4.

Preliminaries
In this section, we give some lemmas which are important for the subsequent sections.
Furthermore, it follows from Lemma 2.1 that is a projector that continuously maps L p,1/ω (R d ) into the subspace V p,1/ω (ϕ).

Lemma 2.4
Let ω, ϕ, andφ be as in Lemma 2.2. Then the function where the modulus of continuity Proof By direct computation, we have A similar computation yields Then Now, we will prove (2.6). In fact, Similarly, we can obtain and Remark 2.1 In fact, V p,1/ω (ϕ) is the range space of P on L p,1/ω (R d ), and For a relatively-separated subset in R d , let U = {β γ } γ ∈ be a bounded uniform partition of unity(BUPU) associated with the covering {B(γ , δ)} γ ∈ , which satisfies Then where V δ is the volume of ball B(γ , δ). Moreover, Then g ∈ V p,1/ω (ϕ).
For any f ∈ L p,1/ω (R d ), we have Now, we will estimate K 2 w . In fact, Similarly, sup x∈R d K 2 (x + ·, x) L 1,ω ≤ r 1 /C ω . Therefore, In the following, we estimate K 3,1 w and K 3,2 w , respectively. In fact, it is easy to verify that Therefore, we obtain Then R A = A R = P and R ,a A ,a = A ,a R ,a = P.
which means that f ∈ V p,1/ω (ϕ) can be exactly recovered.

Asymptotic pointwise error estimation
In this section, we give the asymptotic pointwise error estimates for algorithms (3.3) and (3.4).
Lemma 4.1 Let R be R or R ,a , and K R be its kernel. If δ and a are chosen such that (3.5) or (3.6) is satisfied for algorithm (3.3) or (3.4), respectively, then Proof We only prove (4.1), (4.2) can be proved similarly. Let K R be the kernel of R -P.
Moreover, we have .
Proof Then we obtain Moreover, for each x ∈ R d , This together with (4.3) and (4.4) leads to

Conclusion
In this paper, under a weaker assumption on the generator, we establish the asymptotic pointwise error estimates for reconstructing non-decay shift-invariant signals based on two kinds of average samples. Although we prove the convergence from a theoretical point of view, some numerical experiments are expected to be given for showing the effectiveness of the corresponding iterative reconstruction algorithms, which will be studied in the future work.