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Asymptotic pointwise error estimates for reconstructing shift-invariant signals with generators in a hybrid-norm space
Journal of Inequalities and Applications volume 2021, Article number: 179 (2021)
Abstract
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 \(L_{p,{1 / \omega }}\)-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.
1 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 \(\{f(n\delta )\}_{n\in \mathbf{{Z}}}\) 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 [2–13]. Sampling and reconstruction of signals in a shift-invariant space
is generally studied under the condition that the generator φ is in a Winner amalgam space \(\widetilde{W}_{1,\infty,\omega }(\mathbf {R}^{d})\), which is defined as
Throughout the paper, the weighting function denoted by ω is always assumed to be continuous, symmetric, positive, and submultiplicative,
The decaying weight \(1/\omega \) controls the growing rate of the signals living in \(V_{p,{1 / \omega }}(\varphi )\).
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 \(W_{1,q,\omega }(\mathbf{R}^{d})\) with \(q=\max \{p,p'\}\), \(p'\) is the conjugate number of p. Here, for \(1\leq p,q<\infty \), the weighted hybrid-norm space is defined as
with the norm
If p or q is infinity, usual adjustments are used. Both [11] and [7] gave the error estimates in the sense of \(L_{p,{1 / \omega }}\)-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 \(V_{p,{1 / \omega }}(\varphi )\) under the assumptions
-
(i)
\(\varphi \in W_{1,\infty,\omega }(\mathbf{R}^{d})\).
-
(ii)
\(\lim_{\delta \rightarrow 0}\|\omega _{\delta }(\varphi )\|_{W_{1, \infty,\omega }}=0\), where \(\omega _{\delta }(\varphi )\) is the continuous modulus defined by
$$\begin{aligned} \omega _{\delta }(\varphi ) (x):=\sup_{ \vert y \vert \leq \delta } \bigl\vert \varphi (x+y)- \varphi (x) \bigr\vert . \end{aligned}$$ -
(iii)
There exist positive constants A and B such that
$$\begin{aligned} A\leq \sum_{k\in \mathbf{Z}^{d}} \bigl\vert \widehat{\varphi }(\xi +2k \pi ) \bigr\vert ^{2}\leq B. \end{aligned}$$
In fact, it is easy to verify that assumptions (i)–(ii) are satisfied if \(\varphi \in \widetilde{W}_{1,\infty,\omega }(\mathbf{R}^{d})\) is a continuous function, cf. [4].
The sampling set \(\Gamma \subset \mathbf{R}^{d}\) is assumed to be relatively-separated, that is,
for some \(\delta >0\). Furthermore, \(\delta >0\) is said to be a gap of a relatively-separated subset Γ if
Given a relatively-separated sampling set Γ, two kinds of average sampling schemes are considered. The first one is
where the average sampling functionals \(\{\psi _{\gamma }: \gamma \in \Gamma \}\) satisfy the following:
-
(a)
\(\int _{\mathbf{R}^{d}}\psi _{\gamma }(x)\,dx=1\) for all \(\gamma \in \Gamma \).
-
(b)
There exists \(M>0\) such that \(\int _{\mathbf{R}^{d}}|\psi _{\gamma }(x)|\,dx\leq M\) for all \(\gamma \in \Gamma \).
-
(c)
supp \(\psi _{\gamma }\subset B(\gamma, a) \) 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
where \(\psi \in L_{1}(\mathbf{R}^{d})\) such that \(\int _{\mathbf{R}^{d}}\psi (x)\,dx=1\), \(\psi _{a}(\cdot ):=\frac{1}{a^{d}}\psi (\frac{\cdot }{a})\), and \(\psi _{a}^{\ast }(\cdot )=\overline{\psi _{a}(-\cdot )}\).
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.
2 Preliminaries
In this section, we give some lemmas which are important for the subsequent sections.
Lemma 2.1
([7])
Let ω be a submultiplicative weighting function. If \(\varphi \in W_{1,\infty,\omega }(\mathbf{R}^{d})\) and \(c \in {\ell _{1,\omega }} ( {{{\mathbf{Z}}^{d}}} )\), then the function \(f = \sum_{k \in \mathbf{Z} ^{d}} c_{k}\varphi (\cdot -k)\) also belongs to \(W_{1,\infty,\omega }(\mathbf {R}^{d})\) and
Lemma 2.2
Let ω be a submultiplicative weighting function that additionally satisfies the Gelfand–Raikov–Shilov (GRS) condition
Suppose that the generator φ satisfies assumptions \((i)\)–\((iii)\), then the dual function φ̃ of φ is also in \(W_{1,\infty,\omega }(\mathbf{R}^{d})\) and satisfies \(\mathop{\lim }_{\delta \to 0} { \Vert {{\omega _{\delta }} ( {\tilde{\varphi }} )} \Vert _{{W_{1,\infty,\omega }}}} = 0\).
Proof
By Proposition 6 in [7], \({\tilde{\varphi }} \in W_{1,\infty,\omega }(\mathbf{R}^{d})\). Moreover, there exists a unique \(b \in {\ell _{1,\omega }} ( {{{\mathbf{Z}}^{d}}} )\) such that
Furthermore, it follows from Lemma 2.1 that
Therefore, \(\mathop{\lim }_{\delta \to 0} { \Vert {{\omega _{\delta }} ( {\tilde{\varphi }} )} \Vert _{{W_{1,\infty,\omega }}}} = 0\) is proved. □
Lemma 2.3
([7])
Let ω be a submultiplicative weighting function satisfying the Gelfand–Raikov–Shilov (GRS) condition. Suppose that \({\varphi } \in W_{1,\infty,\omega }(\mathbf{R}^{d})\), then the linear operator
is a projector that continuously maps \(L_{p,{1 / \omega }}(\mathbf{R}^{d})\) into the subspace \(V_{p,{1 / \omega }}(\varphi )\).
Lemma 2.4
Let ω, φ, and φ̃ be as in Lemma 2.2. Then the function
satisfies
and
where the modulus of continuity
Proof
By direct computation, we have
A similar computation yields
Note that \(W_{1,\infty,\omega }(\mathbf{R}^{d})\subseteq W_{1,1,\omega }( \mathbf{R}^{d})= L_{1,\omega }(\mathbf{R}^{d})\). Then
Now, we will prove (2.6). In fact,
Similarly, we can obtain
Note that \(\|f\|_{L_{1,\omega }}\leq \|f\|_{W_{1,\infty,\omega }}\). Then (2.6) follows from Lemma (2.2) and
□
Remark 2.1
In fact, \({V_{p,{1 / \omega }}} ( \varphi )\) is the range space of P on \(L_{p,{1 / \omega }}(\mathbf{R}^{d})\), and
For a relatively-separated subset Γ in \(\mathbf{R}^{d}\), let \(U=\{\beta _{\gamma }\}_{\gamma \in \Gamma }\) be a bounded uniform partition of unity(BUPU) associated with the covering \(\{B(\gamma,\delta )\}_{\gamma \in \Gamma }\), which satisfies
-
(i)
\(0 \leq \beta _{\gamma }(x)\leq 1\);
-
(ii)
\(\beta _{\gamma }\) is supported in \(B(\gamma,\delta )\) for all \(\gamma \in \Gamma \);
-
(iii)
\(\sum_{\gamma \in \Gamma }\beta _{\gamma }(x)\equiv 1, \forall x\in \mathbf{R}^{d}\).
Lemma 2.5
Let \(U=\{\beta _{\gamma }\}_{\gamma \in \Gamma }\) be a BUPU associated with the covering \(\{B(\gamma,\delta )\}_{\gamma \in \Gamma }\). The function \(K(x,y)\) defined by (2.4) satisfies \(\int _{\mathbf{R}^{d}}K(x,y)\beta _{\gamma }(y)\,dy \in V_{p,{1 / \omega }}(\varphi )\).
Proof
Let \(g(x):=\int _{\mathbf{R}^{d}}K(x,y)\beta _{\gamma }(y)\,dy\). Then
where \(V_{\delta }\) is the volume of ball \(B(\gamma,\delta )\). Moreover,
Therefore, \(\|g\|_{L_{p,\frac{1}{\omega }}}\leq \|g\|_{L_{1,\frac{1}{\omega }}}^{ \frac{1}{p}}\|g\|_{L_{\infty,\frac{1}{\omega }}}^{1-\frac{1}{p}}< \infty \), which means that \(g \in L_{p,{1 / \omega }}(\mathbf{R}^{d})\). Furthermore, we have
Then \(g \in V_{p,{1 / \omega }}(\varphi )\). □
Lemma 2.6
Suppose that \(\psi \in L_{1}(\mathbf{R}^{d})\) satisfies \(\lim_{a\rightarrow 0}\int _{|t|\geq 1}|\psi _{a}(t)|\omega (t)\,dt=0\), then the function
satisfies
Proof
Note that
We first estimate \(I_{2}(x,y)\).
Similarly,
Then we have
Since \(\lim_{\delta \rightarrow 0}\|\omega _{\delta }(K)\|_{w}=0\) for any \(\varepsilon > 0\), there exists \(0 < \delta '<1\) such that \(\|\omega _{\delta }(K)\|_{w}<\varepsilon \) for any \(\delta <\delta '\). Then
Moreover, we obtain
and
This together with (2.9)–(2.10) proves that \(\lim_{a\rightarrow 0}\mathop{\sup }_{x \in {{ \mathbf{R}}^{d}}} { \Vert {{I_{1}} ( {x + \cdot,x} )} \Vert _{{L_{1,\omega }}}}=0\). Similarly, \(\lim_{a\rightarrow 0}\mathop{\sup }_{x \in {{ \mathbf{R}}^{d}}} { \Vert {{I_{1}} ( {x,x + \cdot } )} \Vert _{{L_{1,\omega }}}}=0\) and \(\lim_{a\rightarrow 0}\|K_{1}\|_{w}=0\) is proved.
Now, we will prove \(\lim_{a,\delta \rightarrow 0}\|\omega _{\delta }(K_{1})\|_{w}=0\). Note that
In the following, we estimate \(\lim_{a,\delta \rightarrow 0}\|\omega _{\delta }(I_{1})\|_{w}=0\) and \(\lim_{a,\delta \rightarrow 0}\|\omega _{\delta }(I_{2})\|_{w}=0\), respectively. In fact,
and
Therefore, we have
Moreover, \(\lim_{a,\delta \rightarrow 0}\|\omega _{\delta }(I_{1})\|_{w}=0\) follows from \(\lim_{a\rightarrow 0}\|I_{1}\|_{w}=0\) and \(\lim_{\delta \rightarrow 0}\|\omega _{\delta }(K)\|_{w}=0\). By a similar method, we can obtain
Therefore, \(\lim_{a,\delta \rightarrow 0}\|\omega _{\delta }(I_{2})\|_{w}=0\). Finally, \(\lim_{a,\delta \rightarrow 0}\|\omega _{\delta }(K_{1}) \|_{w}=0\) follows from (2.11)–(2.13). □
3 Iterative reconstruction algorithms
For two kinds of average sampling schemes,
define sampling operators for signals in \(V_{p,{1 / \omega }}(\varphi )\) as
Given an initial sequence \(c_{0}= (c_{0}(\gamma ) )_{\gamma \in \Gamma }\in \ell _{p,{1 / \omega }}(\Gamma )\), the corresponding algorithms are formulated as
and
Theorem 3.1
Let \(\Gamma \subset \mathbf{R}^{d}\) be a relatively-separated subset with gap \(\delta >0\) and \(U=\{\beta _{\gamma }\}_{\gamma \in \Gamma }\) be a BUPU associated with the covering \(\{B(\gamma,\delta )\}_{\gamma \in \Gamma }\). If δ and a are chosen such that
then algorithm (3.3) converges to some \(f_{\infty }\in V_{p,1 /\omega }(\varphi )\). If δ and a are chosen such that
then algorithm (3.4) also converges to some \(f_{\infty }\in V_{p,1/ \omega }(\varphi )\). In particular, if \(c_{0}(\gamma )=\langle f,\psi _{\gamma }\rangle \) or \(\langle f,\psi _{a}(\cdot -\gamma )\rangle \) for signals \(f\in V_{p,1 / \omega }(\varphi )\), then \(f_{\infty }=f\). That is, f can be exactly recovered.
Proof
Note that \(f_{n+1}-f_{n}=(P-A_{\Gamma })(f_{n}-f_{n-1})\), \(n\geq 1\), for algorithm (3.3) and \(f_{n+1}-f_{n}=(P-A_{\Gamma,a})(f_{n}-f_{n-1})\), \(n\geq 1\), for algorithm (3.4). It is enough to prove that \(\|P-A_{\Gamma }\|<1\) and \(\|P-A_{\Gamma,a}\|<1\) on \(L_{p,{1/ \omega }}(\mathbf{R}^{d})\).
For any \(f\in L_{p,{1 /\omega }}(\mathbf{R}^{d})\), we have
Now, we will estimate \(\|K_{2}\|_{w}\). In fact,
Similarly, \(\sup_{x\in \mathbf{R}^{d}}\|K_{2}(x+\cdot,x)\|_{L_{1, \omega }}\leq {r_{1}}/{C_{\omega }}\). Therefore,
For any \(f\in L_{p,{1 / \omega }}(\mathbf{R}^{d})\), let \((P-A_{\Gamma,a})f(x):=\int _{\mathbf{R}^{d}}K_{3}(x,y)f(y)\,dy\). Then
In the following, we estimate \(\|K_{3,1}\|_{w}\) and \(\|K_{3,2}\|_{w}\), respectively. In fact, it is easy to verify that
and
Therefore, we obtain
Define operators
and
Then \(R_{\Gamma }A_{\Gamma }=A_{\Gamma }R_{\Gamma }=P\) and \(R_{\Gamma,a}A_{\Gamma,a}=A_{\Gamma,a}R_{\Gamma,a}=P\).
If \(c_{0}(\gamma )=\langle f,\psi _{\gamma }\rangle \) or \(\langle f,\psi _{a}(\cdot -\gamma )\rangle \) for \(f\in V_{p,{1 / \omega }}(\varphi )\), then
or
which means that \(f\in V_{p,{1 / \omega }}(\varphi )\) can be exactly recovered. □
4 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 \(R_{\Gamma }\) or \(R_{\Gamma,a}\), 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
and
Proof
We only prove (4.1), (4.2) can be proved similarly. Let \(K_{\widetilde{R}_{\Gamma }}\) be the kernel of \(R_{\Gamma }-P\). Then \(R_{\Gamma }-P = \sum_{n=1}^{\infty }(P-A_{\Gamma })^{n}\) means that
Moreover, we have
Since \(R_{\Gamma }=(R_{\Gamma }-P)+P\), then
□
Theorem 4.1
Let \(1\leq p\leq \infty \). Suppose that \(\Gamma \subset \mathbf{R}^{d}\) is a relatively-separated subset with gap δ, and \(U=\{\beta _{\gamma }\}_{\gamma \in \Gamma }\) is a BUPU associated with the covering \(\{B(\gamma,\delta )\}_{\gamma \in \Gamma }\). Assume that \(\{\varepsilon (\gamma )\}_{\gamma \in \Gamma }\) is weighted bounded i.i.d. noise with weighted zero mean and \(\sigma ^{2}\) variance, that is,
for some \(B>0\), and that the initial data \(c_{0}\) are \((\langle f,\psi _{\gamma }\rangle +\varepsilon (\gamma ) )_{ \gamma \in \Gamma }\) and \((\langle f,\psi _{a}(\cdot -\gamma )\rangle +\varepsilon ( \gamma ) )_{\gamma \in \Gamma }\) in algorithms (3.3) and (3.4), respectively. Then, for any \(x\in \mathbf{R}^{d}\),
and
Here, \(a, \delta \) satisfy (3.5) for algorithm (3.3), (3.6) for algorithm (3.4), respectively.
Proof
Note that \(f_{\infty }=Rf_{0}\) and \(f=R(f_{0}-h_{0})\), where \(f_{0}=\sum_{\gamma \in \Gamma }c_{0}(\gamma )\int _{ \mathbf{R}^{d}}K(\cdot,y)\beta _{\gamma }(y)\,dy\) and \(h_{0}=\sum_{\gamma \in \Gamma }\varepsilon (\gamma )\int _{ \mathbf{R}^{d}}K(\cdot,y)\beta _{\gamma }(y)\,dy\). Then we have
Note that
Then we obtain
Moreover, for each \(x\in \mathbf{R}^{d}\),
Note that
This together with (4.3) and (4.4) leads to
□
5 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.
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Acknowledgements
The authors thank the referee for his useful suggestions to reform the paper.
Funding
The project is partially supported by the Guangxi Natural Science Foundation (No. 2019GXNSFFA245012), Guangxi Key Laboratory of Cryptography and Information Security (No. GCIS201925), Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation.
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XF carried out the mathematical studies and drafted the manuscript. HZL and YT participated in the design of the study. All authors read and approved the final manuscript.
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Li, H., Fan, X. & Tang, Y. Asymptotic pointwise error estimates for reconstructing shift-invariant signals with generators in a hybrid-norm space. J Inequal Appl 2021, 179 (2021). https://doi.org/10.1186/s13660-021-02712-w
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DOI: https://doi.org/10.1186/s13660-021-02712-w