Pre-wavelet bases in Lebesgue spaces
- Chiou-Yueh Gun^{1},
- Kai-Cheng Wang^{2}Email authorView ORCID ID profile,
- Chi-I Yang^{2} and
- Kuei-Fang Chang^{3}
https://doi.org/10.1186/s13660-015-0811-4
© Gun et al. 2015
Received: 24 April 2015
Accepted: 1 September 2015
Published: 17 September 2015
Abstract
Under the decay condition, we have constructed the dual wavelet basis of a pre-wavelet basis. The frame operators of both bases are bijective on Lebesgue spaces. Both bases are also unconditional bases for Lebesgue spaces.
Keywords
MSC
1 Introduction
This study was motivated by the following two facts. There exists a compactly supported, smooth Riesz wavelet basis such that the corresponding frame operator is bijective on \(L^{2}(\mathbb {R})\), but not on \(L^{p}(\mathbb {R})\) for any specified \(1< p<2\) [1]. However, a frame operator is a composition of a pre-frame operator and the adjoint operator of the pre-frame operator. One therefore wonders whether the Riesz wavelet basis is complete in \(L^{p}(\mathbb {R})\), \(1< p<2\). The second one was found in [2], p.130,136, in which a Riesz wavelet basis with good behavior was found to maybe be not complete in \(L^{p}(\mathbb {R})\), \(2< p<\infty\). A more detailed understanding of such wavelets can be found in Remark 2.2.
This study confirms the bijectivity of the pre-wavelet (semi-orthogonal wavelet) frame operator on \(L^{p}(\mathbb {R})\), \(1< p<\infty\) (Theorem 3.4). Due to bijectivity of the frame operator, it is certain that the pre-frame operator of the wavelet basis is surjective on \(L^{p}(\mathbb {R})\), \(1< p<\infty\). The keys for proving Theorem 3.4 are that the dual wavelet basis of the pre-wavelet basis has been constructed, and it has the same decay as the basis. In contrast, the two Riesz wavelet bases discussed earlier had only put emphasis on the wavelet behavior, but they ignored the behavior of a dual basis. It may yield negative results in the completeness of the wavelet bases in Lebesgue spaces.
This paper is organized as follows. In Section 2, we give the relevant theory of wavelet frames. In Section 3, we prove Lemma 3.1 and Theorems 3.2-3.4. Finally, we give further comments and examples in Section 4.
2 Preliminaries and notations
Theorem 2.1
([3], Theorem 16.1)
- (1)
\(\{f_{n}\}\) is unconditional.
- (2)There exists a positive constant C such that, for all n, for all \(\epsilon_{i}=\pm1\), and for all scalars \(a_{1}, a_{2}, \ldots, a_{n}\),$$\begin{aligned} \Biggl\| {\sum_{i=1}^{n}}\epsilon_{i}a_{i}f_{i} \Biggr\| \leq C \Biggl\| {\sum_{i=1}^{n}}a_{i}f_{i} \Biggr\| . \end{aligned}$$
Remark 2.2
(1) To avoid confusion, we denote by H the frame operator of \(\{S^{-1}\psi_{j,k}\}\). Notice that H is bounded, invertible (\(SH=HS=I\)), self-adjoint, and positive on \(L^{2}(\mathbb {R})\) [6], p.90.
(2) If the frame operator of a frame is bijective on \(L^{p}(\mathbb {R})\), then bijectivity will guarantee the pre-frame operator to be surjective. Therefore, bijectivity guarantees that each function f in \(L^{p}(\mathbb {R})\) has an expansion in terms of the wavelet.
3 Proofs and results
Here, we would like to stress the importance of Lemma 3.1, being the foundation of the main results in this study. The proofs of Lemma 3.1 and Theorem 3.2 are based on the ideas expressed in [11, 12], without the unnecessary assumptions and complicated proving procedures. Most importantly, the results produced in Lemma 3.1 have not been produced in the references mentioned.
Lemma 3.1
Proof of Lemma 3.1
Next, we prove Theorem 3.2, which explains that the frame operator of the Riesz wavelet basis with the decay condition has \(L^{p}\)-boundedness. It has to be stressed that Theorem 3.2 does not need to take the behavior of the dual basis into consideration, and it also does not require the dual basis to have a wavelet structure.
Theorem 3.2
The frame operator S of a Riesz wavelet basis \(\mathcal{F_{\psi}}\) is of weak type \((1,1)\) and of type \((p,p)\), for all \(1< p<\infty\).
Proof of Theorem 3.2
Theorem 3.3 gives a characterization on a pair of biorthogonal Riesz wavelet bases for Lebesgue spaces. The proof relies on the \(L^{p}\)-boundedness of the frame operator of the dual wavelet basis which is an application from Theorem 3.2.
Theorem 3.3
- (1)The operator H is of weak type \((1,1)\) and of type \((p,p)\), for all \(1< p<\infty\). Moreover,for all \(f\in L^{p}(\mathbb {R})\) and some constants \(C^{\prime}\), \(C^{\prime\prime}\), \(C_{1}^{\prime}\), \(C_{1}^{\prime\prime}\).$$\begin{aligned} &C^{\prime}\|f\|_{p}\leq\|Sf\|_{p}\leq C^{\prime\prime} \|f\|_{p},\\ &C_{1}^{\prime}\|f\|_{p}\leq\| Hf\|_{p}\leq C_{1}^{\prime\prime}\|f\|_{p}, \end{aligned}$$
- (2)
The two operators S and H are bijective on \(L^{p}(\mathbb {R})\), \(1< p<\infty\).
- (3)
\(\mathcal{F_{\psi}}\) and \(\mathcal{F_{\widetilde{\psi}}}\) are also unconditional bases for \(L^{p}(\mathbb {R})\), \(1< p<\infty\).
Proof of Theorem 3.3
(3) In order to prove (3), we will change hypotheses (3.1)-(3.5) to:
Theorem 3.4
- (1)The dual basis \(\{\widetilde{\psi}_{j,k}:j,k\in \mathbb {Z}\}\) of \(\{ \psi_{j,k}:j,k\in \mathbb {Z}\}\) has a wavelet structure, and the generator ψ̃ satisfies$$\begin{aligned} \|\sigma\widetilde{\psi}\|_{w}\leq\|\sigma\psi\|_{w}\| \widetilde {\psi}\|_{\mathcal{A}}< \infty. \end{aligned}$$
- (2)
The two operators S and H are bijective on \(L^{p}(\mathbb {R})\), \(1< p<\infty\). Both \(\{\psi_{j,k}:j,k\in \mathbb {Z}\}\) and \(\{\widetilde{\psi }_{j,k}:j,k\in \mathbb {Z}\}\) are also unconditional bases for \(L^{p}(\mathbb {R})\), \(1< p<\infty\).
Proof of Theorem 3.4
(2) We have constructed the dual basis of \(\{\psi_{j,k}\}\) in (1), and \(\{\widetilde{\psi}_{j,k}\}\) has the same decay as \(\{\psi_{j,k}\}\). Applying that and Theorem 3.3, we complete the proof of (2). □
4 Further remarks and examples
In this section, we give comments and examples. Bownik and Weber point out the connection between the behavior of a canonical dual of \(\{\psi _{j,k}\}\) and how ψ generates a GMRA [15]. For a Riesz wavelet ψ, the shift invariance of negative dilates \(V_{0}(\psi)\) implies a wavelet structure of the canonical dual. They have also given an interesting example of a wavelet frame for which the canonical dual does not have the wavelet structure, but other dual frames with the wavelet structure exist. Bownik and Lemvig also mentioned in another paper [8] that a frame may have an infinite number of dual frames with a wavelet structure. The last paper [16] we refer to was written by Lemvig; he constructed pairs of dual band-limited wavelet frames.
The aforementioned three papers all depend on one important theorem [6], p.277, Theorem 12.1.3 (or [17], p.263, [16], Theorem 2.1). Part of the results are generated by using the multiresolution analysis (MRA). However, Theorem 3.4 provides a method to construct the dual wavelet of a given pre-wavelet without using the mentioned theorem or MRA. Through the fact that dual basis has wavelet structure and it has the same decay as basis, less limitations are applied on wavelets to accomplish our results.
Next, we review some typical pre-wavelets and biorthogonal wavelets.
(I) Compactly supported. Compactly supported wavelets certainly satisfy condition \(\mathcal{M}\) and thus Theorem 3.3 can be applied to all compactly supported orthogonal wavelets. The earliest example is the Haar wavelet [18], Chapter 1, Example A. For non-orthogonal cases, Chui and Wang [19], and independently Jia and Micchelli [14] have constructed compactly supported pre-wavelets. Riemenschneider considered a cardinal spline approach to construct compactly supported, skew-symmetric pre-wavelets [20, 21]. Cohen et al. have constructed biorthogonal wavelets that are symmetric, regular, and compactly supported [7].
Unser has constructed pre-wavelets in [28] that these wavelets converge to a cosine-modulated Gaussian function as the degree of the spline goes to infinity. Another remarkable wavelet was found in [29] which has subexponential decay (\(\psi(x)\leq E_{\gamma} e^{-| x |^{1-\gamma}}\), \(0<\gamma<1\), \(x\in \mathbb {R}\)), it is band-limited and belongs to \(C^{\infty}\).
For different methods to construct dual wavelets, Kim has shown that if a Riesz wavelet is associated with an MRA, then it has a dual Riesz wavelet [30], Corollary 2.9. Later, he gave a characterization on biorthogonal/semi-orthogonal wavelets associated with MRA [31]. Combining these two facts, lots of biorthogonal/semi-orthogonal wavelets can be found.
(III) Smooth. Daubechies’ wavelets and biorthogonal wavelets in [7] are typical smooth wavelets. References [32], p.856, Theorem 3.3, [33, 34], [35], p.295, Theorems 9.1.5-9.1.6, [18], Chapter 5, Theorems 6.14, 6.23, [2], Chapter 6, and [36], Section 7.3, Theorem 1, ensure that orthogonal wavelets that are sufficiently smooth are unconditional bases for Lebesgue spaces and the associated frame operators have \(L^{p}\)-boundedness. Bui and Laugesen have proven that the frame operator of a wavelet frame (not necessarily a wavelet basis) is bijective on Lebesgue spaces if some assumptions are provided [37]. Mainly ψ and \(\psi'\) need to have sufficient decay on the frequency domain. The aforementioned results were obtained by using the technique of Calderón-Zygmund operators. Apparently, the method suffers from the requirement of smoothness on wavelets (and thus they do not support the Haar wavelet). In contrast, smoothness is not required for Theorems 3.2-3.4 to hold.
Declarations
Acknowledgements
The many suggestions and detailed corrections of anonymous referees are gratefully acknowledged. The authors thank language editor Weilling Chen who has made a significant revision of the manuscript.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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