Regular and Irregular Sampling Theorem for Multiwavelet Subspaces

Abstract

We study the sampling theorem for frames in multiwavelet subspaces. Firstly, a sufficient condition under which the regular sampling theorem holds is established. Then, notice that irregular sampling is also useful in practice; we consider the general cases of the irregular sampling and establish a general irregular sampling theorem for multiwavelet subspaces. Finally, using this generalized irregular sampling theorem, we obtain an estimate for the perturbations of regular sampling in shift-invariant spaces.

1. Introduction

At the present time the sampling theorem plays a crucial role in signal processing and communication, as it establishes an equivalence between discrete signals and analogue (continuous) signals. For a band-limited signal, the classical Shannon sampling theorem provides an exact representation by its uniform samples with a sampling rate higher than its Nyquist rate. But there exist several problems. Firstly, real-world signals or images are never exactly band-limited. Secondly, there is no such device as an ideal (antialiasing or reconstruction) low-pass filter. Thirdly, Shannon reconstruction formula is rarely used in practice (especially with images) because of the slow decay of the sinc function. Therefore, this classical Shannon sampling theorem has been generalized to many other forms.

Extensions of Shannon sampling theorem to scalar wavelets can be found in [1–5], but a scalar wavelet cannot have the orthogonality, compact support, and symmetry at the same time (except the Haar wavelet). It is a disadvantage for signal processing. Meanwhile, multiwavelets have attracted much attention in the research community, since multiwavelet has more desired properties than any scalar wavelet function, such as orthogonality, short compact support, symmetry, high approximation order, and so on. The first orthogonal multiwavelet with symmetry, approximation order, and compact support was presented by Geronimo et al. [6]. In addition, the sampling theorems for multiwavelet subspaces were studied in [7–10]. The authors of [7, 9] presented the construction of compactly supported orthogonal multiscaling functions that are continuously differentiable and cardinal. The scaling functions thereby support a Shannon-like sampling theorem. However, the multiwavelets of [7, 9] do not have symmetry. It is not good for digital signal processing and image compression. They also did not study the sampling theorem for frame in multiwavelet subspaces, which is very important in application.

A reconstruction from more general sets of points is necessary if the measurements cannot be made at uniform points. Hence, irregular sampling is also useful in practice. In [11], the authors obtained a Feichtinger-Grochenig iterative algorithm based on the quasiinterpolation projection procedure to recover signals from irregular samples for multiwavelet subspaces. However, the maximal allowable gap between two sampling points needed for reconstructing a function from its samples was not obtained, which was supposed to exist in theory. The authors of [10] generalized the multiwavelet sampling theorem by reproducing a kernel and derived an estimate for the perturbations of uniform noninteger sampling in shift-invariant spaces, but their results just based on the Riesz basis.

In our paper, we will show a sufficient condition for regular sampling theorem to hold in multiwavelet subspaces for frames. Notice that a reconstruction from more general sets of points is necessary if the measurements cannot be made an uniform; we establish the irregular sampling theorems in multiwavelet subspaces. Finally, an estimate for the perturbations of regular sampling in shift-invariant spaces is derived.

This paper is organized as follows. Section 2 contains some definitions in this correspondence. Also, we review some relative notations. In Section 3, we discuss general uniform noninteger sampling and obtain a sufficient condition for uniform noninteger sampling theorem to hold. In Section 4, an irregular sampling theorem in general multiwavelet subspaces is established. Finally, by applying the result in Section 4, we estimate the perturbations of uniform noninteger sampling in shift-invariant spaces.

2. Preliminary

We now introduce some notations used in this correspondence.

𝐶(𝑅) is the space of continuous function.

The shift-invariant closed subspace 𝑉₀ generated by {𝜙₁,𝜙₂,…,𝜙_𝑟}𝑉₀=span{𝜙_𝑖(⋅−𝑘)∶1≤𝑖≤𝑟,𝑘∈𝑍}⊂𝐿²(𝑅). (2.1)

For a function 𝑓∈𝐿²(𝑅), we denote by ⟨𝑓⟩ the minimal closed shift invariant subspace that contains 𝑓.

Let 𝐟=[𝑓₁,𝑓₂,…,𝑓_𝑟]^𝑇 (2.2) denote vector (we denote vectors and matrices in this paper in boldface). The integration ∫_𝑅𝐟(𝑥)𝑑𝑥 is defined as _𝑅𝐟(𝑥)𝑑𝑥=[_𝑅𝑓₁(𝑥)𝑑𝑥,_𝑅𝑓₂(𝑥)𝑑𝑥,…,_𝑅𝑓_𝑟(𝑥)𝑑𝑥]^𝑇. (2.3)

The Fourier transform of vector 𝐟 is defined by 𝐟(̂𝜔)=_𝑅𝐟(𝑥)𝑒^{−𝑖𝑥𝜔}𝑑𝑥. (2.4)

The inverse Fourier transform of vector 𝐟 is written by 𝐟(̌𝑥)=12𝜋_𝑅𝐟(𝜔)𝑒^𝑖𝑥𝜔𝑑𝜔. (2.5)

𝑍_𝑓(𝑥,𝜔)=∑_𝑛∈𝑍𝑓(𝑥+𝑛)𝑒^{−𝑖𝑛𝜔} is the Zak transform of function 𝑓. The Zak transform of vector 𝐟 is defined by 𝑍_{𝐟(𝑥,𝜔)}=[𝑍_𝑓1(𝑥,𝜔),𝑍_𝑓2(𝑥,𝜔),…,𝑍_𝑓𝑟(𝑥,𝜔)]^𝑇. (2.6)

A collection of elements {𝜙_𝑖(⋅−𝑘)∶1≤𝑖≤𝑟,𝑘∈𝑍} in a Hilbert space 𝐻 is called a frame if there exist constants 𝐴 and 𝐵, 0<𝐴≤𝐵<∞, such that 𝐴‖𝑓‖²≤^𝑟_{𝑘∈𝑍𝑖=1}|⟨𝑓,𝜙_𝑖(⋅−𝑘)⟩|²≤𝐵‖𝑓‖²,∀𝑓∈𝐻. (2.7)

If {𝜙_𝑖(⋅−𝑘)∶1≤𝑖≤𝑟,𝑘∈𝑍} is a frame for 𝐻, then there exists a dual frame {𝜙_𝑘}_𝑘∈𝑍 for {𝜙_𝑖(⋅−𝑘)∶1≤𝑖≤𝑟,𝑘∈𝑍} [12, Theorem 5.6.5].

For 𝑓∈𝐻,we can write𝐸_𝑓={𝜔∈𝑅∣𝐺_𝑓𝐺(𝜔)>0},_𝑓(𝜔)=_𝑘∈𝑍|𝑓(𝜔+2𝜋𝑘)|²,𝜒_𝐸𝑓=1,𝑡∈𝐸_𝑓,0,𝑡∉𝐸_𝑓. (2.8)

LetΦ=[𝜙₁,𝜙₂,…,𝜙_𝑟]^𝑇,Φ=[𝜙₁,𝜙₂,…,𝜙_𝑟]^𝑇. (2.9) The reproducing kernel is defined as 𝑞(𝑡,⋅)=^𝑟_{𝑙=1𝑘∈𝑍}𝜙_𝑙(𝑡−𝑘)𝜙_𝑙(⋅−𝑘)=_𝑘∈𝑍Φ^∗(𝑡−𝑘)Φ(⋅−𝑘), (2.10) where ∗ denotes the Hermitian conjugate. Put 𝑡=𝑛+𝑎_𝑚. Then 𝑞(𝑛+𝑎_𝑚,⋅)=^𝑟_{𝑙=1𝑘∈𝑍}𝜙_𝑙(𝑛+𝑎_𝑚−𝑘)𝜙_𝑙(⋅−𝑘)=_𝑘∈𝑍Φ^∗(𝑛+𝑎_𝑚−𝑘)Φ(⋅−𝑘), (2.11) where 1≤𝑚≤𝑟, 𝑛∈𝑍, 𝑎_𝑚∈[0,1) is constant, and 𝑎_𝑖≠𝑎_𝑗 for 𝑖≠𝑗, 𝑖,𝑗=1,2,…,𝑟.

Let 𝐴⊂𝑅, Max{𝐴} means the largest number in subset 𝐴, and Min{𝐴} means the smallest number in subset 𝐴.

For 𝑓,𝑔∈𝐿²(𝑅), let bracket function [𝑓,𝑔] be the function defined a.e. by [𝑓,𝑔](𝜉)=∑_𝑘∈𝑍𝑓(𝜉+𝑘)𝑔(𝜉+𝑘).

3. General Uniform Sampling Theorem

The main purpose of this section is to study the regular sampling theorem for frame in multiwavelet subspace.

Firstly, we start with some useful lemmas.

Lemma 3.1. Let 𝑓,𝑔∈𝐿²(𝑅). If ⟨𝑓⟩⟂⟨𝑔⟩ and |ℎ|,|𝑠|∈⟨𝑓⟩⊕⟨𝑔⟩ for all 𝑠∈⟨𝑓⟩,ℎ∈⟨𝑔⟩, then [𝑠,ℎ]=0𝑎.𝑒.

Proof. Suppose that {𝑓(⋅−𝑘)} is a frame sequence in 𝐿²(𝑅). Let 𝑠∈⟨𝑓⟩, then there exists a sequence {𝑐_𝑖}_𝑖∈𝑍∈𝑙² such that 𝑠=∑_𝑖∈𝑍𝑐_𝑖𝑓(⋅−𝑖).
Define𝜒(𝑠)=1,𝑠≥0,−1,𝑠<0. (3.1) We have |𝑠|=𝜒(𝑠)𝑠=𝜒(𝑠)_𝑖∈𝑍𝑐_𝑖𝑓(⋅−𝑖)=_𝑖∈𝑍𝜒(𝑠)𝑐_𝑖𝑓(⋅−𝑖). (3.2) It is easy to check that {𝜒(𝑠)𝑐_𝑖}_𝑖∈𝑍∈𝑙² and |𝑠|⟂⟨𝑔⟩; thus |𝑠|∈⟨𝑓⟩. Similar to the above argument, forallℎ∈⟨𝑔⟩, we get |ℎ|∈⟨𝑔⟩.
Suppose that ⟨𝑓⟩⟂⟨𝑔⟩ holds; by |𝑠|∈⟨𝑓⟩ and |ℎ|∈⟨𝑔⟩, we can obtain0≤_𝑇|_𝑘∈𝑍𝑠(𝑥+𝑘)⋅≤ℎ(𝑥+𝑘)|𝑑𝑥_{𝑇𝑘∈𝑍}|𝑠(𝑥+𝑘)⋅=ℎ(𝑥+𝑘)|𝑑𝑥_𝑅|𝑠(𝑥)⋅ℎ(𝑥)|𝑑𝑥≤⟨|𝑠|,|ℎ|⟩=0. (3.3) It implies that [𝑠,ℎ]=0𝑎.𝑒. The proof is completed.

Lemma 3.2. Let {𝑓_𝑖}_{𝑖∈{1,2,…,𝑟}}⊂𝐿²(𝑅), then the following two assertions are equivalent:(a)for any {𝑐^𝑖_𝑘}_𝑘∈𝑍∈𝑙², 𝑖∈{1,2,…,𝑟}, ∑^𝑟_𝑖=1∑_𝑘∈𝑍𝑐^𝑖_𝑘𝑓_𝑖(𝑥−𝑘) converges pointwise to a continuous function;(b){f_i}_{i∈{1,2,…,r}}⊂C(R) and sup_𝑥∑^𝑟_𝑖=1∑_𝑘∈𝑍|𝑓_𝑖(𝑥−𝑘)|²<+∞.

Proof. (a)⇒(b). It is easy to see that forall𝑖∈{1,2,…,𝑟},𝑓_𝑖∈𝐶(𝑅). For each 𝑥∈𝑅, since ∑^𝑟_𝑖=1∑_𝑘∈𝑍𝑐^𝑖_𝑘𝑓_𝑖(𝑥−𝑘) is convergent for each {𝑐^𝑖_𝑘}∈𝑙², clearly, ∑_𝑘∈𝑍|𝑓_𝑖(𝑥−𝑘)|²<∞. For each 𝑥∈[0,1], define𝑇_𝑥𝑐=^𝑟_{𝑖=1𝑘∈𝑍}𝑐^𝑖_𝑘𝑓_𝑖(𝑥−𝑘),∀𝑐={𝑐^𝑖_𝑘}∈𝑙². (3.4) Then 𝑇_𝑥 is a bounded linear functional on 𝑙² with the norm ‖𝑇_𝑥‖=(∑^𝑟_𝑖=1∑_𝑘∈𝑍|𝑓_𝑖(𝑥−𝑘)|²)^1/2. For any {𝑐^𝑖_𝑘}∈𝑙², define 𝑔(𝑡)=∑^𝑟_𝑖=1∑_𝑘∈𝑍𝑐^𝑖_𝑘𝑓_𝑖(𝑡−𝑘). Since 𝑔(𝑡) is continuous on 𝑅, we have sup_𝑥∈[0,1]|𝑇_𝑥𝑐|=sup_𝑥∈[0,1]|𝑔(𝑥)|<+∞. (3.5) By the Banach-Steinhaus theorem [13], sup_𝑥∈[0,1]‖𝑇_𝑥‖<+∞, that is, ∑^𝑟_𝑖=1∑_𝑘∈𝑍|𝑓_𝑖(𝑥−𝑘)|² is bounded on 𝑅.
(a)⇒(b). By the Cauchy inequality, ∑^𝑟_𝑖=1∑_𝑘∈𝑍𝑐^𝑖_𝑘𝑓_𝑖(𝑥−𝑘) in convergent uniformly on 𝑅, so the limit function is continuous.

Lemma 3.3. Let {𝜙_𝑖(⋅−𝑘)∶1≤𝑖≤𝑟,𝑘∈𝑍} be a frame for 𝑉₀ with bounds A and B. If ⟨𝜙_𝑖⟩⟂⟨𝜙_𝑗⟩ for all 𝑖,𝑗∈{1,2,…,𝑟}, 𝑖≠𝑗, then {𝜙_𝑖(⋅−𝑘)}_𝑘∈𝑍 is a frame for the subspace ⟨𝜙_𝑖⟩.

Proof. Suppose that {𝜙_𝑖(⋅−𝑘)∶1≤𝑖≤𝑟,𝑘∈𝑍} is a frame for the subspace 𝑉₀, then there exist constants 0<𝐴≤𝐵<∞ such that 𝐴‖𝑓‖²≤^𝑟_{𝑖=1𝑘∈𝑍}|⟨𝑓,𝜙_𝑖(𝑥−𝑘)⟩|²≤B‖𝑓‖². (3.6) Notice that ⟨𝜙_𝑖⟩⟂⟨𝜙_𝑗⟩ for all 𝑖,𝑗∈{1,2,…,𝑟}, 𝑖≠𝑗; if 𝑓∈⟨𝜙_𝑖⟩, then ⟨𝑓,𝜙_𝑗(𝑥−𝑘)⟩=0. Hence, 𝐴‖𝑓‖²≤_𝑘∈𝑍|⟨𝑓,𝜙_𝑖(𝑥−𝑘)⟩|²≤B‖𝑓‖². (3.7) By the definition of Frame, {𝜙_𝑖(⋅−𝑘)}_𝑘∈𝑍 is a frame for the closed subspace ⟨𝜙_𝑖⟩. Then we get the desired result.

Lemma 3.4. Let 𝑓∈𝐿²(𝑅) and 𝑓∈𝐶(𝑅), then there exists a set 𝐸∈𝑅, |𝑅⧵𝐸|=0 such that for any 𝑥∈𝐸 and forall{𝑐_𝑘}_𝑘∈𝑍∈𝑙², ∑_𝑘∈𝑍𝑐_𝑘𝑓(𝑥−𝑘) converges pointwise to a continuous function.

Proof. By 𝑓∈𝐿²(𝑅), then ∫_𝑇∑_𝑘∈𝑍|𝑓(𝑥−𝑘)|²𝑑𝑥<∞. Hence there exists a set 𝐸∈𝑅,|𝑅⧵𝐸|=0 such that sup_𝑥∈𝐸∑_𝑘∈𝑍|𝑓(𝑥−𝑘)|²<∞. Notice that 𝑓∈𝐶(𝑅); using Lemma 3.2, clearly, for any {𝑐_𝑘}_𝑘∈𝑍∈𝑙², ∑_𝑘𝑐_𝑘𝑓(𝑥−𝑘) converges pointwise to a continuous function in set 𝐸.

From the above lemmas, we have the following result.

Theorem 3.5. Let {𝜙_𝑖(⋅−𝑘)∶1≤𝑖≤𝑟,𝑘∈𝑍} be a frame for 𝑉₀. Suppose that 𝜙_𝑖, 1≤𝑖≤𝑟 are continuous functions, ⟨𝜙_𝑖⟩⟂⟨𝜙_𝑗⟩ for all 𝑖≠𝑗, 𝑖,𝑗∈{1,2,…,𝑟}, and sup^𝑟_{𝑥∈𝑅𝑖=1𝑘∈𝑍}|𝜙_𝑖(𝑥−𝑘)|²<+∞. (3.8) Then 𝑓(𝑡)=_𝑅𝑓(𝑠)𝑞(𝑡,𝑠)𝑑𝑠 (3.9) holds for each 𝑓∈𝑉₀ and 𝑡∈𝑅.

Proof. Suppose that {𝜙_𝑖(⋅−𝑘)∶1≤𝑖≤𝑟,𝑘∈𝑍} is a frame for 𝑉₀ and ⟨𝜙_𝑖⟩⟂⟨𝜙_𝑗⟩, 𝑖≠𝑗, then, by [14, Proposition 3.1] and Lemma 3.3, there exist constants 0<𝐴_𝑖≤𝐵_𝑖<∞ such that𝐴_𝑖𝜒_𝐸𝜙𝑖(𝜔)≤_𝑛∈𝑍|𝜙_𝑖(𝜔+2𝑛𝜋)|²≤𝐵_𝑖𝜒_𝐸𝜙𝑖(𝜔),𝑎.𝑒. (3.10) From Lemma 4 in [15], we have the dual frame 𝜙_𝑖 for {𝜙_𝑖(⋅−𝑘)∶1≤𝑖≤𝑟,𝑘∈𝑍}, where 𝜙_𝑖 is defined by 𝜙_𝑖𝜙(𝜔)=^𝑖(𝜔)𝐺_𝜙𝑖(𝜔),𝜔∈𝐸_𝜙𝑖,0,𝜔∉𝐸_𝜙𝑖. (3.11)
Hence, for all 1≤𝑖≤𝑟,𝜙_𝑖(𝜔)𝐵_𝑖≤𝜙_𝑖𝜙(𝜔)≤^𝑖(𝜔)𝐴_𝑖 (3.12) holds.
So, it follows that𝜙_𝑖(𝑥)=_𝜔∈𝑅𝜙_𝑖(𝜔)𝑒^𝑖𝜔𝑥1𝑑𝜔≤𝐴_{𝑖𝜔∈𝑅}𝜙_𝑖(𝜔)𝑒^𝑖𝜔𝑥1𝑑𝜔=𝐴_𝑖𝜙_𝑖𝜙(𝑥),_𝑖(𝑥)=_𝜔∈𝑅𝜙_𝑖(𝜔)𝑒^𝑖𝜔𝑥1𝑑𝜔≥𝐵_{𝑖𝜔∈𝑅}𝜙_𝑖(𝜔)𝑒^𝑖𝜔𝑥1𝑑𝜔=𝐵_𝑖𝜙_𝑖(𝑥). (3.13)
From above results, obviously, ifsup^𝑟_{𝑥∈𝑅𝑖=1𝑘∈𝑍}|𝜙_𝑖(𝑥−𝑘)|²<+∞, (3.14) then sup^𝑟_{𝑥∈𝑅𝑖=1𝑘∈𝑍}|𝜙_𝑖(𝑥−𝑘)|²<+∞. (3.15) Hence, 𝑞(𝑡,𝑠)=^𝑟_{𝑙=1𝑘∈𝑍}𝜙_𝑙(𝑡−𝑘),𝜙_𝑙(𝑠−𝑘)=_𝑘∈𝑍Φ^∗(𝑡−𝑘)Φ(𝑠−𝑘) (3.16) converges to 𝑞(𝑡,⋅) in 𝐿²(𝑅) for any 𝑓∈𝑉₀ and 𝑡∈𝑅.
By ⟨𝜙_𝑖⟩⟂⟨𝜙_𝑗⟩ for all 𝑖,𝑗∈{1,2,…,𝑟}, 𝑖≠𝑗, it is easy to check that {𝜙_𝑖(⋅−𝑘)∶1≤𝑖≤𝑟,𝑘∈𝑍} is a dual frame for {𝜙_𝑖(⋅−𝑘)∶1≤𝑖≤𝑟,𝑘∈𝑍}; then we have𝑓(𝑡)=^𝑟_{𝑙=1𝑘∈𝑍}⟨𝑓(𝑠),𝜙_𝑙(𝑠−𝑘)⟩𝜙_𝑙(𝑡−𝑘)=⟨𝑓(𝑠),^𝑟_{𝑙=1𝑘∈𝑍}𝜙_𝑙(𝑡−𝑘),𝜙_𝑙(𝑠−𝑘)⟩=⟨𝑓(𝑠),_𝑘∈𝑍Φ^∗=(𝑡−𝑘)Φ(𝑠−𝑘)⟩_𝑅𝑓(𝑠)𝑞(𝑡,𝑠)𝑑𝑠 (3.17) for any 𝑓∈𝑉₀ and 𝑡∈𝑅. So we get the desired result.

Based on these facts, the following sampling theorem is established.

Theorem 3.6. Let {𝜙_𝑖(⋅−𝑘)∶1≤𝑖≤𝑟,𝑘∈𝑍} be a frame for 𝑉₀. Suppose that ⟨𝜙_𝑖⟩⟂⟨𝜙_𝑗⟩ for all 𝑖,𝑗∈{1,2,…,𝑟}, 𝑖≠𝑗, and 𝜙_𝑖, 1≤𝑖≤𝑟 are continuous such that sup_t∈R^r_i=1k∈Z|𝜙_i(t−k)|²<+∞. (3.18) If there exist constants 𝐴_𝑖 and 𝐵_𝑖, 0<𝐴_𝑖≤𝐵_𝑖<+∞ such that 𝐴_𝑖𝜒_𝐸𝜙𝑖(𝜔)≤|𝜙^∗_𝑖(𝜔)|≤𝐵_𝑖𝜒_𝐸𝜙𝑖(𝜔),𝑎.𝑒.𝜔∈𝑅, (3.19) where 𝜙^∗_𝑖(𝜔)=∑_𝑛∈𝑍𝜙_𝑖(𝑛)𝑒^{−𝑖𝑛𝜔}, then there exists a frame {𝑠_𝑖(⋅−𝑘)∶1≤𝑖≤𝑟,𝑘∈𝑍} of 𝑉₀, for 𝑖,𝑗∈{1,2,…,𝑟}, 𝑖≠𝑗𝑓(𝑡)=^𝑟_{𝑛∈𝑍𝑚=1}𝑓(𝑛+𝑎_𝑚)𝑠_𝑚(𝑡−𝑛),𝑓∈𝑉₀, (3.20) holds, where the convergence is in 𝐿²(𝑅).

Proof. Define the function 𝑞_𝑖(𝑡,𝑠) by 𝑞_𝑖(𝑡,𝑠)=∑_𝑘∈𝑍𝜙_𝑖(𝑡−𝑘)𝜙_𝑖(𝑠−𝑘), where 𝜙_𝑖 is defined by (3.11). From Lemma 3.3, it is easy to see that {𝜙_𝑖(⋅−𝑘)}_𝑘∈𝑍 and {𝜙_𝑖(⋅−𝑘)}_𝑘∈𝑍 are the frames for the subspace ⟨𝜙_𝑖⟩.
Notice that𝑞_𝑖(𝑛,𝜔)=𝜙^∗_𝑖(𝜔)𝜙_𝑖𝐴(𝜔),_𝑖𝜒_𝐸𝜙𝑖(𝜔)≤|𝜙^∗_𝑖(𝜔)|≤𝐵_𝑖𝜒_𝐸𝜙𝑖(𝜔),𝑎.𝑒.𝜔∈𝑅. (3.21) From the proof process of Proposition 3.1 in [14], {𝑞_𝑖(𝑛,⋅)}_𝑛∈𝑍={𝑞_𝑖(0,⋅−𝑛)}_𝑛∈𝑍 is a frame for the subspace ⟨𝜙_𝑖⟩. Similar to the above argument, obviously, for 1≤𝑚≤𝑟, {𝑞_𝑖(𝑛+𝑎_𝑚,⋅)}_𝑛∈𝑍={𝑞_𝑖(0,⋅−𝑛−𝑎_𝑚)}_𝑛∈𝑍 is the frame for the subspace ⟨𝜙_𝑖⟩. Hence, forall𝑓_𝑖∈𝑉_𝑖,forall𝑚∈{1,2,…,𝑟}, there exist constants 0<𝐶^𝑚_𝑖≤𝐷^𝑚_𝑖<∞ such that 𝐶^𝑚_𝑖‖𝑓_𝑖‖²≤_𝑛∈𝑍|⟨𝑓_𝑖,𝑞_𝑖(𝑛+𝑎_𝑚,⋅)⟩|²≤𝐷^𝑚_𝑖‖𝑓_𝑖‖². (3.22) Thus, we have 𝑓_𝑖(𝑠)=_𝑛∈𝑍⟨𝑓_𝑖(𝑠),𝑞_𝑖(𝑛+𝑎_𝑚,𝑠)⟩𝑞_𝑖(𝑛+𝑎_𝑚,𝑠), (3.23) where 𝑞_𝑖(𝑛+𝑎_𝑚,𝑠) is the dual frame for 𝑞_𝑖(𝑛+𝑎_𝑚,𝑠).
From above results, we get^𝑟_𝑖=1𝐶^𝑚_𝑖‖𝑓_𝑖‖²≤^𝑟_{𝑖=1𝑛∈𝑍}|⟨𝑓_𝑖,𝑞_𝑖(𝑛+𝑎_𝑚,𝑠)⟩|²≤^𝑟_𝑖=1𝐷^𝑚_𝑖‖𝑓_𝑖‖², (3.24) where 𝑚∈{1,2,…,𝑟}.
Notice that {𝜙_𝑖(⋅−𝑘)}_𝑘∈𝑍, {𝑞_𝑖(0,⋅−𝑛−𝑎_𝑚)}_{𝑛∈𝑍,𝑚∈{1,2,…,𝑟}} and {𝑞_𝑖(0,⋅−𝑛)}_𝑛∈𝑍 are the frame for subspace ⟨𝜙_𝑖⟩; by [16, Lemma 3], we have 𝐸_𝜙𝑖=𝐸_{𝑞𝑖(0,⋅−𝑛−𝑎𝑚)}=𝐸_{𝑞𝑖(0,⋅−𝑛)} (except a null measurable set).
By Lemma 3.3, it follows that for all 𝑓∈𝑉₀, 𝑓=∑^𝑟_𝑖=1𝑓_𝑖 holds, where 𝑓_𝑖∈𝑉_𝑖. Again by Lemma 3.3, we have 𝑉_𝑖⟂𝑉_𝑙, 1≤𝑖≠𝑙≤𝑟, then,‖𝑓‖²=⟨^𝑟_𝑖=1𝑓_𝑖,^𝑟_𝑙=1𝑓_𝑙⟩=^𝑟𝑟_𝑖=1𝑙=1⟨𝑓_𝑖,𝑓_𝑙⟩=^𝑟_𝑖=1⟨𝑓_𝑖,𝑓_𝑖⟩=^𝑟_𝑖=1‖𝑓_𝑖‖². (3.25) From (3.23) and (3.25), forall𝑚∈{1,2,…,𝑟}, we get Min{𝐶^𝑚_𝑖}^𝑟_𝑖=1‖𝑓‖²=Min{𝐶^𝑚_𝑖}^𝑟𝑟_𝑖=1𝑖=1‖𝑓_𝑖‖²≤^𝑟_𝑖=1𝐶^𝑚_𝑖‖𝑓_𝑖‖²≤^𝑟_{𝑖=1𝑛∈𝑍}|⟨𝑓_𝑖,𝑞_𝑖(𝑛+𝑎_𝑚,)⟩|²,_{𝑟𝑖=1𝑛∈𝑍}|⟨𝑓_𝑖,𝑞_𝑖(𝑛+𝑎_𝑚,⋅)⟩|²≤^𝑟_𝑖=1𝐷^𝑚_𝑖‖𝑓_𝑖‖²≤Max{𝐷^𝑚_𝑖}^𝑟𝑟_𝑖=1𝑖=1‖𝑓_𝑖‖²=Max{𝐷^𝑚_𝑖}^𝑟_𝑖=1‖𝑓‖². (3.26)
For ⟨𝑓(𝑡),𝑞_𝑖(𝑛+𝑎_𝑖,𝑡)⟩=_𝑅^𝑟_𝑙=1𝑓_𝑙(𝑡)𝑞_𝑖(𝑛+𝑎_𝑖=,𝑡)𝑑𝑡_{𝑟𝑙=1𝑅}𝑓_𝑙(𝑡)𝑞_𝑖(𝑛+𝑎_𝑖=,𝑡)𝑑𝑡_𝑅𝑓_𝑖(𝑡)𝑞_𝑖(𝑛+𝑎_𝑖,𝑡)𝑑𝑡=⟨𝑓_𝑖(𝑡),𝑞_𝑖(𝑛+𝑎_𝑖,𝑡)⟩, (3.27) notice (3.26), we have Min{𝐶^𝑖_𝑖}^𝑟_𝑖=1‖𝑓‖²=Min{𝐶^𝑖_𝑖}^𝑟𝑟_𝑖=1𝑖=1‖𝑓_𝑖‖²≤^𝑟_𝑖=1𝐶^𝑖_𝑖‖𝑓_𝑖‖²≤^𝑟_{𝑛∈𝑍𝑖=1}|⟨𝑓_𝑖,𝑞_𝑖(𝑛+𝑎_𝑖,⋅)⟩|²=^𝑟_{𝑛∈𝑍𝑖=1}|⟨𝑓,𝑞_𝑖(𝑛+𝑎_𝑖,⋅)⟩|²,_{𝑟𝑛∈𝑍𝑖=1}|⟨𝑓,𝑞_𝑖(𝑛+𝑎_𝑖,⋅)⟩|²=^𝑟_{𝑛∈𝑍𝑖=1}|⟨𝑓_𝑖,𝑞_𝑖(𝑛+𝑎_𝑖,⋅)⟩|²≤^𝑟_𝑖=1𝐷^𝑖_𝑖‖𝑓_𝑖‖²≤Max{𝐷^𝑖_𝑖}^𝑟𝑟_𝑖=1𝑖=1‖𝑓_𝑖‖²=Max{𝐷^𝑖_𝑖}^𝑟_𝑖=1‖𝑓‖². (3.28)
Let 𝐶=Min{𝐶^𝑖_𝑖}^𝑟_𝑖=1,𝐷=Max{𝐷^𝑖_𝑖}^𝑟_𝑖=1, then forall𝑓∈𝑉₀, there exist constants number 0<𝐶≤𝐷<∞ such that𝐶‖𝑓‖²≤^𝑟_{𝑖=1𝑛∈𝑍}|⟨𝑓,𝑞_𝑖(𝑛+𝑎_𝑖,⋅)⟩|²≤𝐷‖𝑓‖². (3.29) Hence, according to the definition of frame, {𝑞_𝑖(𝑛+𝑎_𝑖,𝑠)∶1≤𝑖≤𝑟,𝑛∈𝑍} is the frame for the subspace 𝑉₀. Let 𝑞(𝑛+𝑎_𝑖,𝑠)=∑^𝑟_𝑙=1𝑞_𝑙(𝑛+𝑎_𝑖,𝑠); by Theorem 3.5, clearly, ⟨𝑓(𝑠),𝑞(𝑛+𝑎_𝑖,𝑠)⟩=𝑓(𝑛+𝑎_𝑖).
Notice that|⟨𝑓(𝑠),^𝑟_𝑙=1𝑞_𝑙(𝑛+𝑎_𝑖,𝑠)⟩|²=|^𝑟_𝑙=1⟨𝑓(𝑠),𝑞_𝑙(𝑛+𝑎_𝑖,𝑠)⟩|²≤^𝑟_𝑙=1|⟨𝑓(𝑠),𝑞_𝑙(𝑛+𝑎_𝑖,𝑠)⟩|². (3.30) Using (3.27) and (3.22), then ^r_{𝑖=1𝑛∈𝑍}|⟨𝑓(𝑠),𝑞(𝑛+𝑎_𝑖,𝑠)⟩|²=^𝑟_{𝑖=1𝑛∈𝑍}|⟨𝑓(𝑠),^𝑟_𝑙=1𝑞_𝑙(𝑛+𝑎_𝑖,𝑠)⟩|²≤^𝑟_𝑖=1^𝑟_{𝑛∈𝑍𝑙=1}|⟨𝑓(𝑠),𝑞_𝑙(𝑛+𝑎_𝑖,𝑠)⟩|²=^𝑟_𝑖=1^𝑟_{𝑛∈𝑍𝑙=1}|⟨𝑓_𝑙(𝑠),𝑞_𝑙(𝑛+𝑎_𝑖,s)⟩|²≤^𝑟_𝑙=1(^𝑟_𝑖=1𝐷^𝑖_𝑙‖𝑓_𝑙‖²)≤^𝑟_𝑙=1Max{𝐷^𝑖_𝑙∶𝑖∈{1,2,…,𝑟}}‖𝑓_𝑙‖²≤Max{Max{𝐷^𝑖_𝑙∶𝑖∈{1,2,…,𝑟}}∶𝑙∈{1,2,…,𝑟}}‖𝑓‖²=𝐷^′‖𝑓‖². (3.31)
For 𝑓_𝑙∈⟨𝜙_𝑙⟩,𝑓_𝑝∈⟨𝜙_𝑝⟩, by Lemmas 3.1 and 3.4, notice that for all 𝑙∈{1,2,…,𝑟}, {𝑓_𝑙(𝑛+𝑎_𝑖)}_𝑛∈𝑍∈𝑙², then ∑_𝑛∈𝑍𝑓_𝑙(𝑛+𝑎_𝑖)𝑓_𝑝(𝑛+𝑎_𝑖)=0, for all 𝑖∈{1,2,…,𝑟}, 𝑙≠𝑝. Hence,^𝑟_{𝑖=1𝑛∈𝑍}|⟨𝑓,𝑞(𝑛+𝑎_𝑖,⋅)⟩|²=^𝑟_{𝑖=1𝑛∈𝑍}(⟨𝑓,𝑞(𝑛+𝑎_𝑖,⋅)⟩)(⟨𝑓,𝑞(𝑛+𝑎_𝑖=,⋅)⟩)_{𝑟𝑖=1𝑛∈𝑍}(^𝑟_𝑙=1⟨𝑓_𝑙,𝑞_𝑙(𝑛+𝑎_𝑖,⋅)⟩)(^𝑟_𝑝=1⟨𝑓_𝑝,𝑞_𝑝(𝑛+𝑎_𝑖=,⋅)⟩)_{𝑟𝑖=1𝑟𝑛∈𝑍𝑙=1}|⟨𝑓_𝑙,𝑞_𝑙(𝑛+𝑎_𝑖,⋅)⟩|²≥Min{Min{𝐶^𝑖_𝑙∶𝑖∈{1,2,…,𝑟}}∶𝑙∈{1,2,…,𝑟}}‖𝑓‖²≥𝐶^′‖𝑓‖² (3.32) holds. From (3.31) and (3.32), we get that {𝑞(𝑛+𝑎_𝑖,𝑠)∶𝑖=1,2,…,𝑟,𝑛∈𝑍} is the frame for the subspace 𝑉₀. Then there exists a dual frame {𝑠_𝑖(⋅−𝑘)∶1≤𝑖≤𝑟,𝑘∈𝑍} of {𝑞(𝑛+𝑎_𝑖,𝑠)∶𝑖=1,2,…,𝑟,𝑛∈𝑍} such that 𝑓=^𝑟_{𝑖=1𝑛∈𝑍}⟨𝑓,𝑞(𝑛+𝑎_𝑖,⋅)⟩𝑠_𝑖(⋅−𝑘)=^𝑟_{𝑖=1𝑛∈𝑍}𝑓(𝑎_𝑖+𝑛)𝑠_𝑖(⋅−𝑘),∀𝑓∈𝑉₀. (3.33) The proof is completed.

4. General Irregular Sampling Theorem

In many realistic situations, we try to reconstruct a continuous signal 𝑓∈𝑉₀ by using its discrete samples {𝑓(𝑛+𝑎_𝑚+𝛿^𝑛_𝑚)∶1≤𝑚≤𝑟,𝑛∈𝑍}. Obviously, the samples cannot be arbitrary; we must discuss the values 𝛿^𝑛_𝑚 of deviation from the uniformly noninteger points. The following theorem studies this problem.

Based on the results in Section 3, we get the irregular sampling theorems.

Theorem 4.1. Let {𝜙_𝑖(⋅−𝑘)∶1≤𝑖≤𝑟,𝑘∈𝑍} be a frame for 𝑉₀ with bounds A and B. Suppose that ⟨𝜙_𝑖⟩⟂⟨𝜙_𝑗⟩ for all 𝑖,𝑗∈{1,2,…,𝑟}, 𝑖≠𝑗, and 𝜙_𝑖, 1≤𝑖≤𝑟 are continuous such that sup^𝑟_{𝑡∈𝑅𝑖=1𝑘∈𝑍}|𝜙_𝑖(𝑡−𝑘)|²<+∞. (4.1) Moreover, suppose that there exist constants 𝐴_𝑖 and 𝐵_𝑖, 0<𝐴_𝑖≤𝐵_𝑖<+∞ such that 𝐴_𝑖𝜒_𝐸𝜙𝑖(𝜔)≤|𝜙^∗_𝑖(𝜔)|≤𝐵_𝑖𝜒_𝐸𝜙𝑖(𝜔),𝑎.𝑒.𝜔∈𝑅. (4.2) If there exists a sequence {𝛿^𝑛_𝑚∶n∈𝑍,𝑚=1,2,…,𝑟} and two constants 𝐶 and 𝐷, 0<𝐶≤𝐷<∞ such that 𝐶^𝑟_{𝑚=1𝑛∈𝑍}|𝑓(𝑛+𝑎_𝑚)|²≤^𝑟_{𝑚=1𝑛∈𝑍}|𝑓(𝑛+𝑎_𝑚+𝛿^𝑛_𝑚)−𝑓(𝑛+𝑎_𝑚)|²≤𝐷^𝑟_{𝑚=1𝑛∈𝑍}|𝑓(𝑛+𝑎_𝑚)|² (4.3) holds for any 𝑓∈𝑉₀. Then, there exists a frame {𝑠_𝑚,𝑛(𝑡)∶1≤𝑚≤𝑟,𝑛∈𝑍} for 𝑉₀ such that for any 𝑓∈𝑉₀, 𝑓(𝑡)=^𝑟_{𝑛∈𝑍𝑚=1}𝑓(𝑛+𝑎_𝑚+𝛿^𝑛_𝑚)𝑠_𝑚,𝑛(𝑡) (4.4) holds, where the convergence is both in 𝐿²(𝑅) and uniform on R.

Proof. From the argument of Theorem 3.6, it is easy to see that {𝑞(𝑎₁,⋅−𝑛),𝑞(𝑎₂,⋅−𝑛),…,𝑞(𝑎_𝑟,⋅−𝑛)}_𝑛∈𝑍={𝑞(𝑛+𝑎₁,⋅),𝑞(𝑛+𝑎₂,⋅),…,𝑞(𝑛+𝑎_𝑟,⋅)}_𝑛∈𝑍 is a frame for the closed subspace 𝑉₀ of 𝐿²(𝑅). Hence, there exist two constants 𝐶₁ and 𝐶₂, 0<𝐶₁≤𝐶₂<∞ such that 𝐶₁‖𝑓‖²≤^𝑟_{𝑛∈𝑍𝑘=𝑙}|⟨𝑓,𝑞(𝑛+𝑎_𝑘,⋅)⟩|²=^𝑟_{𝑛∈𝑍𝑘=𝑙}|𝑓(𝑛+𝑎_𝑘)|²≤𝐶₂‖𝑓‖²,∀𝑓∈𝑉₀. (4.5)
By (4.3), we have𝐶𝐶₁‖𝑓‖²≤^𝑟_{𝑛∈𝑍𝑘=𝑙}|⟨𝑓,𝑞(𝑛+𝑎_𝑘+𝛿^𝑛_𝑘,⋅)⟩|²=^𝑟_{𝑛∈𝑍𝑘=𝑙}|𝑓(𝑛+𝑎_𝑘+𝛿^𝑛_𝑘)|²≤𝐷𝐶₂‖𝑓‖². (4.6) Then,{𝑞(𝑛+𝑎₁+𝛿^𝑛₁,⋅),𝑞(𝑛+𝑎₂+𝛿^𝑛₂,⋅),…,𝑞(𝑛+𝑎_𝑟+𝛿^𝑛_𝑟,⋅)}_𝑛 is also a frame for the closed subspace 𝑉₀ of 𝐿²(𝑅). So, there exists a dual frame {𝑠_𝑖(⋅−𝑘)∶1≤𝑖≤𝑟,𝑘∈𝑍} of {𝑞(𝑛+𝑎_𝑖+𝛿^𝑛_𝑖,⋅)∶1≤𝑖≤𝑟,𝑘∈𝑍} such that 𝑓(𝑡)=^𝑟_{𝑛∈𝑍𝑚=1}𝑓(𝑛+𝑎_𝑚+𝛿^𝑛_𝑚)𝑠_𝑚(𝑡−𝑛) (4.7) for any 𝑓∈𝑉₀.

Theorem 4.2. Let {𝜙_𝑖(⋅−𝑘)∶1≤𝑖≤𝑟,𝑘∈𝑍} be a frame for 𝑉₀ with bounds A and B. Suppose that ⟨𝜙_𝑖⟩⟂⟨𝜙_𝑗⟩ for all 𝑖,𝑗∈{1,2,…,𝑟}, 𝑖≠𝑗, and 𝜙_𝑖, 1≤𝑖≤𝑟 are continuous such that sup^𝑟_{𝑡∈𝑅𝑖=1𝑘∈𝑍}|𝜙_𝑖(𝑡−𝑘)|²<+∞. (4.8) Moreover, suppose that there exist constants 𝐴_𝑖 and 𝐵_𝑖, 0<𝐴_𝑖≤𝐵_𝑖<+∞ such that 𝐴_𝑖𝜒_𝐸𝜙𝑖(𝜔)≤|𝜙^∗_𝑖(𝜔)|≤𝐵_𝑖𝜒_𝐸𝜙𝑖(𝜔),𝑎.𝑒.𝜔∈𝑅. (4.9) If there exists a sequence {𝛿^𝑛_𝑚∶𝑛∈𝑍,𝑚=1,2,…,𝑟} and a constant 0<𝜃<1 such that ^𝑟_{𝑚=1𝑛∈𝑍}|𝑓(𝑛+𝑎_𝑚+𝛿^𝑛_𝑚)−𝑓(𝑛+𝑎_𝑚)|²≤𝜃^2r_{𝑚=1𝑛∈𝑍}|𝑓(𝑛+𝑎_𝑚)|² (4.10) holds for any 𝑓∈𝑉₀, then, there exists a frame {𝑠_𝑚,𝑛(𝑡)∶1≤𝑚≤𝑟,𝑛∈𝑍} for 𝑉₀ such that for any 𝑓∈𝑉₀𝑓(𝑡)=^𝑟_{𝑛∈𝑍𝑚=1}𝑓(𝑛+𝑎_𝑚+𝛿^𝑛_𝑚)𝑠_𝑚,𝑛(𝑡) (4.11) holds, where the convergence is both in 𝐿²(𝑅) and uniform on R.

Proof. If ^𝑟_{𝑚=1𝑛∈𝑍}|𝑓(𝑛+𝑎_𝑚+𝛿^𝑛_𝑚)−𝑓(𝑛+𝑎_𝑚)|²≤𝜃^2𝑟_{𝑚=1𝑛∈𝑍}|𝑓(𝑛+𝑎_𝑚)|², (4.12) then (1−𝜃²)^𝑟_{𝑚=1𝑛∈𝑍}|𝑓(𝑛+𝑎_𝑚)|²≤^𝑟_{𝑚=1𝑛∈𝑍}|𝑓(𝑛+𝑎_𝑚+𝛿^𝑛_𝑚)|²≤(1+𝜃²)^𝑟_{𝑚=1𝑛∈𝑍}|𝑓(𝑛+𝑎_𝑚)|². (4.13) By Theorem 4.1, we get the desired result.

5. Perturbation of Uniform Noninteger Sampling in Shift-Invariant Spaces

Our objective is to find explicit formulas or algorithms to calculate the ranges of the perturbations of uniform noninteger sampling in shift-invariant spaces. In order to establish the algorithm for perturbations of uniform noninteger sampling for frame in shift-invariant spaces, we need to introduce the function class 𝐿^𝜆_𝑎𝑚[𝑎,𝑏](𝜆>0,𝜎∈[0,1),0∈[𝑎,𝑏]⊂[−1,1]) defined and used in [10].

Definition 5.1. 𝐿^𝜆_𝑎𝑚[𝑎,𝑏](𝜆>0 and 𝜎∈[0,1),0∈[𝑎,𝑏]⊂[−1,1]) consists of all the measurable functions f, for which the norm ‖𝑓‖_{𝐿𝜆𝑎𝑚[𝑎,𝑏]}=sup_{{𝑟𝑘}𝑘⊂[𝑎,𝑏]}∑_𝑘|𝑓(𝑘+𝜎+𝑟_𝑘)−𝑓(𝑘+𝜎)|sup_𝑘|𝑟_𝑘|^𝜆<∞. (5.1)

Theorem 5.2. Let {𝜙_𝑖(⋅−𝑘)∶1≤𝑖≤𝑟,𝑘∈𝑍} be a frame for 𝑉₀ with bounds A and B. Suppose that there exist constants 𝐴_𝑖 and 𝐵_𝑖, 0<𝐴_𝑖≤𝐵_𝑖<+∞ such that 𝐴_𝑖𝜒_𝐸𝜙𝑖(𝜔)≤|𝜙^∗_𝑖(𝜔)|≤𝐵_𝑖𝜒_𝐸𝜙𝑖(𝜔),𝑎.𝑒.𝜔∈𝑅. (5.2) Moreover, suppose that ⟨𝜙_𝑖⟩⟂⟨𝜙_𝑗⟩ for all 𝑖,𝑗∈{1,2,…,𝑟}, 𝑖≠𝑗, and 𝜙_𝑖, 1≤𝑖≤𝑟 are continuous such that sup^𝑟_{𝑡∈𝑅𝑖=1𝑘∈𝑍}|𝜙_𝑖(𝑡−𝑘)|²<+∞. (5.3) Then, for any {𝛿^𝑛_𝑚}⊂[−𝜃_Φ,𝜃_Φ]∩[𝑎,𝑏], 1≤𝑚≤𝑟, 𝑛∈𝑍, there is a frame {𝑆_𝑚,𝑛} of 𝑉₀ such that 𝑓(𝑡)=^𝑟_{𝑛∈𝑍𝑚=1}𝑓(𝑛+𝑎_𝑚+𝛿^𝑛_𝑚)𝑠_𝑚(𝑡−𝑛) (5.4) holds in 𝐿²(𝑅) for any 𝑓∈𝑉₀ if 𝛿_Φ𝐶<(^1/2∑^𝑟_𝑚=1‖𝜙_𝑚‖_{𝐿𝜆𝑎𝑚[𝑎,𝑏]})^1/𝜆. (5.5)

Proof. Notice that {𝜙_𝑖(⋅−𝑘)∶1≤𝑖≤𝑟,𝑘∈𝑍} is a frame for 𝑉₀, then, forall𝑓∈𝑉₀, there exists a square summable sequence {𝑐^𝑛_𝑚} such that 𝑓=∑_𝑛∈𝑍∑^𝑟_𝑚=1𝑐^𝑛_𝑚𝜙_𝑚(⋅−𝑛) holds in 𝐿²(𝑅).
Similar to the argument of Theorem 5 in [10], letΔ=^𝑟_{𝑛∈𝑍𝑚=1}|𝑓(𝑛+𝑎_𝑚+𝛿^𝑛_𝑚)−𝑓(𝑛+𝑎_𝑚)|²=^𝑟_{𝑛∈𝑍𝑚=1}|^𝑟_{𝑘∈𝑍𝑖=1}𝑐^𝑘_𝑖(𝜙_𝑖(𝑛+𝑎_𝑚+𝛿^𝑛_𝑚−𝑘)−𝜙_𝑖(𝑛+𝑎_𝑚−𝑘))|²=^𝑟_{𝑛∈𝑍𝑚=1}^𝑟_{𝑘∈𝑍𝑖=1}^𝑟_{𝑙∈𝑍𝑗=1}𝑐^𝑘_𝑖𝑐^𝑙_𝑗(𝜙_𝑖(𝑛+𝑎_𝑚+𝛿^𝑛_𝑚−𝑘)−𝜙_𝑖(𝑛+𝑎_𝑚−𝑘))×(𝜙_𝑗(𝑛+𝑎_𝑚+𝛿^𝑛_𝑚−𝑘)−𝜙_𝑗(𝑛+𝑎_𝑚−𝑘))=^𝑟_{𝑘∈𝑍𝑖=1}^𝑟_{𝑙∈𝑍𝑗=1}𝑐^𝑘_𝑖𝑐^𝑙_𝑗^𝑟_{𝑛∈𝑍𝑚=1}(𝜙_𝑖(𝑛+𝑎_𝑚+𝛿^𝑛_𝑚−𝑘)−𝜙_𝑖(𝑛+𝑎_𝑚−𝑘))×(𝜙_𝑗(𝑛+𝑎_𝑚+𝛿^𝑛_𝑚−𝑘)−𝜙_𝑗(𝑛+𝑎_𝑚−𝑘)). (5.6) Take 𝑎_{𝑘,𝑖;𝑙,𝑗}=^𝑟_{𝑛∈𝑍𝑚=1}(𝜙_𝑖(𝑛+𝑎_𝑚+𝛿^𝑛_𝑚−𝑘)−𝜙_𝑖(𝑛+𝑎_𝑚−𝑘))×(𝜙_𝑗(𝑛+𝑎_𝑚+𝛿^𝑛_𝑚−𝑘)−𝜙_𝑗(𝑛+𝑎_𝑚−𝑘)). (5.7) Then, 𝑎_{𝑘,𝑖;𝑙,𝑗}=𝑎_{𝑙,𝑗;𝑘,𝑖} holds for any 𝑘,𝑙∈𝑍, 1≤𝑖,𝑗≤𝑟, and Δ=^𝑟_{𝑘∈𝑍𝑖=1}^𝑟_{𝑙∈𝑍𝑗=1}𝑐^𝑘_𝑖𝑐^𝑙_𝑗𝑎_{𝑘,𝑖;𝑙,𝑗}≤^𝑟_{𝑘∈𝑍𝑖=1}^𝑟_{𝑙∈𝑍𝑗=1}|𝑎_{𝑘,𝑖;𝑙,𝑗}|((𝑐^𝑘_𝑖)²+(𝑐^𝑙_𝑗)²)2=12(^𝑟_{𝑘∈𝑍𝑖=1}(^𝑟_{𝑙∈𝑍𝑗=1}|𝑎_{𝑘,𝑖;𝑙,𝑗}|)(𝑐^𝑘_𝑖)²+^𝑟_{𝑙∈𝑍𝑗=1}(^𝑟_{𝑘∈𝑍𝑖=1}|𝑎_{𝑘,𝑖;𝑙,𝑗}|)(𝑐^𝑙_𝑗)²)=^𝑟_{𝑘∈𝑍𝑖=1}(^𝑟_{𝑙∈𝑍𝑗=1}|𝑎_{𝑘,𝑖;𝑙,𝑗}|)(𝑐^𝑘_𝑖)²≤(sup_{𝑘∈𝑍,1≤𝑖≤𝑟}^𝑟_{𝑙∈𝑍𝑗=1}|𝑎_{𝑘,𝑖;𝑙,𝑗}|)^𝑟_{𝑘∈𝑍𝑖=1}(𝑐^𝑘_𝑖)². (5.8) Moreover sup_{𝑘∈𝑍,1≤𝑖≤𝑟}^𝑟_{𝑙∈𝑍𝑗=1}|𝑎_{𝑘,𝑖;𝑙,𝑗}|≤sup_{𝑘∈𝑍,1≤𝑖≤𝑟}^𝑟_{𝑙∈𝑍𝑗=1}^𝑟_{𝑛∈𝑍𝑚=1}|𝜙_𝑖(𝑛+𝑎_𝑚+𝛿^𝑛_𝑚−𝑘)−𝜙_𝑖(𝑛+𝑎_𝑚−𝑘)|×|𝜙_𝑗(𝑛+𝑎_𝑚+𝛿^𝑛_𝑚−𝑙)−𝜙_𝑗(𝑛+𝑎_𝑚−𝑙)|=sup_{𝑘∈𝑍,1≤𝑖≤𝑟}^𝑟_{𝑛∈𝑍𝑚=1}|𝜙_𝑖(𝑛+𝑎_𝑚+𝛿^𝑛_𝑚−𝑘)−𝜙_𝑖(𝑛+𝑎_𝑚−𝑘)|×^𝑟_{𝑙∈𝑍𝑗=1}|𝜙_𝑗(𝑛+𝑎_𝑚+𝛿^𝑛_𝑚−𝑙)−𝜙_𝑗(𝑛+𝑎_𝑚−𝑙)|≤(^𝑟_𝑚=1‖𝜙_𝑚‖_{𝐿𝜆𝑎𝑚[𝑎,𝑏]})²(sup_𝑛,𝑚|𝛿^𝑛_𝑚|²). (5.9) Hence Δ≤^𝑟_{𝑘∈𝑍𝑖=1}(𝑐^𝑘_𝑖)²(^𝑟_𝑚=1‖𝜙_𝑚‖_{𝐿𝜆𝑎𝑚[𝑎,𝑏]})²(sup_𝑛,𝑚|𝛿^𝑛_𝑚|^𝜆)². (5.10)
From the argument in Theorem 3.6, {𝑞(𝑛+𝑎_𝑖,⋅)∶𝑛∈𝑍,1≤𝑖≤𝑟} is a frame for the closed subspace 𝑉₀. By (3.25), then^𝑟_{𝑖=1𝑛∈𝑍}|𝑓(𝑛+𝑎_𝑖)|²=^𝑟_{𝑖=1𝑛∈𝑍}|⟨𝑓,𝑞(𝑛+𝑎_𝑖)⟩|²≥𝐶‖𝑓‖²≥𝐶^𝑟_𝑖=1‖𝑓‖_𝑖. (5.11) Again from the proof process in Theorem 3.6, obviously, ‖𝑓_𝑖‖²≥_𝑘∈𝑍|⟨𝑓_𝑖,𝑞_𝑖(𝑘+𝑎_𝑖,⋅)⟩|²=_𝑘∈𝑍|𝑓_𝑖(𝑘)|²=12𝜋‖_{𝑘∈𝑍𝑙∈𝑍}𝑐^𝑙_𝑖𝜙_𝑖(𝑘−𝑙)𝑒^{−𝑖𝑘𝜔}‖²_∗=1‖𝜙2𝜋^∗𝑖_𝑘∈𝑍𝑐^𝑘_𝑖‖²_∗≥𝐴²_{𝑖𝑘∈𝑍}(𝑐^𝑘_𝑖)² (5.12) holds, where ‖𝑓‖_∗ is defined by ‖𝑓‖_∗=(₀^2𝜋|𝑓(𝑡)|²)^1/2, (5.13) then ^𝑟_{𝑖=1𝑛∈𝑍}|𝑓(𝑛+𝑎_𝑖)|²≥𝐶Min{𝐴²₁,𝐴²₂,…,𝐴²_𝑟}^𝑟_{𝑖=1𝑘∈𝑍}(𝑐^𝑘_𝑖)². (5.14) By Theorem 4.2, we only need to show that (^𝑟_𝑚=1‖𝜙_𝑚‖_{𝐿𝜆𝑎𝑚[𝑎,𝑏]})²(sup_𝑛,𝑚|𝛿^𝑛_𝑚|^𝜆)²≤𝜃²𝐶. (5.15) This is exactly implied by (5.5). We get the desired result.

6. Conclusion

Multiwavelets have orthogonality, regularity, short compact support, symmetry, and high approximation order. It is not possible in the scalar wavelet case. In our paper, we study the sampling theorem for frames in multiwavelet subspaces and obtain the sufficient condition under which the regular sampling theorem holds. Notice that the measurements may not be made at uniform points; we establish an irregular sampling theorem for frame in multiwavelet subspaces. Finally, an estimate for the perturbations of regular sampling in shift-invariant spaces is derived.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (no. 10971228) and the Henan Provincial Natural Science Foundation of China (no. 102300410205).