# Pre-wavelet bases in Lebesgue spaces

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## 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

bijectivity Calderón-Zygmund decomposition theorem frame Riesz basis semi-orthogoal unconditional basis wavelet Wiener’s lemma### MSC

42C40 42C15## 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

*π*of \({\Bbb{N}}\) (or \({\Bbb{Z}}\)). It is well known that unconditional bases can be characterized as follows.

### Theorem 2.1

([3], Theorem 16.1)

*Let*\(\{f_{n}\}\)

*be a basis in a Banach space*.

*Then the following properties are equivalent*.

- (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*\(a_{1}, a_{2}, \ldots, a_{n}\),$$\begin{aligned} \Biggl\| {\sum_{i=1}^{n}}\epsilon_{i}a_{i}f_{i} \Biggr\| \leq C \Biggl\| {\sum_{i=1}^{n}}a_{i}f_{i} \Biggr\| . \end{aligned}$$

*f*belongs to \(W(L^{\infty}, l^{1})\) if it satisfies

*Q*be a mapping from \(L^{p}({\Bbb{R}})\) to \(L^{q}({\Bbb{R}})\), for all \(1\leq p, q\leq\infty\). Then

*Q*is of type \((p,q)\) [5] if

*A*does not depend on

*f*. Similarly,

*Q*is of weak type \((p,q)\) if

*A*does not depend on

*f*or

*α*, and

*m*is the Lebesgue measure. We note that an operator of type \((p,q)\) is also of weak type \((p,q)\).

*A*,

*B*are called frame bounds and if \(A=B\), we call this a tight frame. Frames are essentially a set of functions that span the whole space but are not necessarily linearly independent. Define the frame operator

*S*of \(\{\psi_{i}:i\in \mathbb {Z}\}\),

*T*, which is bounded; it is called the pre-frame operator or the synthesis operator. We have

*S*and \(S^{-1}\) are of type \((2,2)\), bounded, invertible, self-adjoint, and positive [6], p.90. \(\{S^{-1}\psi _{i}:i\in \mathbb {Z}\}\) is also a frame in \({\mathcal {H}}\) and its frame operator is \(S^{-1}\). The canonical dual frame \(\{S^{-1}\psi_{i}:i\in \mathbb {Z}\}\) of a tight frame \(\{\psi_{i}:i\in \mathbb {Z}\}\) is simply \(\{\frac{1}{A}\psi _{i}:i\in \mathbb {Z}\}\).

*ψ*is defined as \(\{\psi_{j,k}:j,k\in \mathbb {Z}\}\), and it is a frame for \(L^{2}(\mathbb {R})\). One can show that

### 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.

*θ*by

*θ̃*such that \(\{\theta_{j,k}\}\) and \(\{\widetilde {\theta}_{j,k}\}\) are a pair of biorthogonal Riesz wavelet bases does not exist. However, \(S^{-1}\theta\) is not always compactly supported even when

*θ*has compact support [6], p.276. Even when

*θ*belongs to the Schwartz class, \(S^{-1}\theta_{j,0}\) does not belong to \(L^{p}\) for small \(p-1>0\) [9]. Moreover, Tao pointed out that the frame operator of \(\{\theta_{j,k}\}\) with ‘good’ conditions does not necessarily come with the bijectivity [1]. This leads to the observation that continuously strengthening restrictions on the wavelets may not necessarily provide effectively guaranteed behavior of dual bases.

*ψ*satisfies condition \(\mathcal{M}\) if

*ψ*, and \(\mathcal{F_{\psi}}:=\{\psi_{j,k}:\psi\in\mathcal {M}, j, k\in{\Bbb{Z}}\}\) is a Riesz basis for \(L^{2}{(\mathbb {R})}\) with \(\psi \in\mathcal{M}\).

## 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

*Let*

*ψ*

*satisfy condition*\(\mathcal{M}\).

*Then*\(\psi\in W(L^{\infty}, l^{1})\),

*and*(3.6)

*holds*.

*Moreover*,

*the following are equivalent*:

*The finiteness does not depend on*

*m*

*or*

*n*.

### Proof of Lemma 3.1

*k*into parts. First,

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

*S*can be proven if

*S*being of weak type \((1,1)\) can be proven. We note that

*S*is of type \((2,2)\). The proof is based on two inequalities:

*f*or

*α*.

*S*being of weak type \((1,1)\) can be proven by the following:

*S*is of type \((p,p)\), \(1< p<2\), and thus by duality for all

*p*, \(2< p<\infty\).

*m*or

*n*, such that

*N*derives from:

*p*, to

*p*,

*j*, and

*k*,

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

*Let*\(\mathcal{F_{\psi}}\)

*and*\(\mathcal{F_{\widetilde{\psi}}}\)

*be a pair of biorthogonal Riesz wavelet bases for*\(L^{2}(\mathbb {R})\).

*Then*:

- (1)
*The operator**H**is of weak type*\((1,1)\)*and of type*\((p,p)\),*for all*\(1< p<\infty\).*Moreover*,$$\begin{aligned} &C^{\prime}\|f\|_{p}\leq\|Sf\|_{p}\leq C^{\prime\prime} \|f\|_{p},\\ &C_{1}^{\prime}\|f\|_{p}\leq\| Hf\|_{p}\leq C_{1}^{\prime\prime}\|f\|_{p}, \end{aligned}$$*for all*\(f\in L^{p}(\mathbb {R})\)*and some constants*\(C^{\prime}\), \(C^{\prime\prime}\), \(C_{1}^{\prime}\), \(C_{1}^{\prime\prime}\). - (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

*H*can be proven. Next, we will prove that

*S*is bounded below. By duality, for all \(f^{\prime}\in L^{2}\cap L^{p}(\mathbb {R})\), we have

*f*in \(L^{p}\)-norm,

*S*is injective and has a closed range. Similarly, we can prove that

*H*is bounded below by using the method described above.

*S*is bijective, that the adjoint of

*S*is bounded below needs to be proven. We denote by \(S^{*}\) the adjoint of

*S*which is on \(L^{q}(\mathbb {R})\) into \(L^{q}(\mathbb {R})\), and which is based on (3.13),

*S*is bijective on \(L^{p}(\mathbb {R})\), \(1< p<\infty\). Similarly, we can prove that

*H*is bijective on \(L^{p}(\mathbb {R})\), \(1< p<\infty \) by using the method described above.

(3) In order to prove (3), we will change hypotheses (3.1)-(3.5) to:

*Q*on \(L^{2}(\mathbb {R})\) as

*Q*can be proven. Secondly, since \(L^{2}\cap L^{p}({\Bbb{R}})\) is a dense subset of \(L^{p}({\Bbb{R}})\), \(1< p<\infty\), we have \(f= {\sum_{j,k\in \mathbb {Z}}\langle f,\widetilde{\psi}_{j,k}\rangle\psi_{j,k}\in L^{p}({\Bbb{R}})}\), for any \(f\in L^{2}\cap L^{p}({\Bbb{R}})\). Thus, for each \(f\in L^{p}({\Bbb{R}})\), the series \({\sum_{j,k\in \mathbb {Z}}\langle f,\widetilde{\psi }_{j,k}\rangle\psi_{j,k}}\) converges unconditionally (since

*Q*is of type \((p,p)\)) in \(L^{p}({\Bbb{R}})\) so that it must converge to

*f*. Similarly, we can prove that \(\mathcal{F_{\widetilde{\psi}}}\) is an unconditional basis of \(L^{p}(\mathbb {R})\), \(1< p<\infty\) by using the methods described above. □

*σ*is the weight function which satisfies \(\sigma (x+y)\leq\sigma(x)\sigma(y)\). For example, \(\sigma(x)=(1+x)^{p}\), \(p>1\). So, if \(\{\sigma(k)a_{k}\}\in l^{1}(\mathbb {Z})\),

*f*is a continuous function of

*z*on the torus \(\mathbb{T}\), and \(\mathcal{A}\) is a commutative Banach algebra with the unit. In our application, we denote

### Theorem 3.4

*Let*

*ψ*

*be a pre*-

*wavelet of*\(L^{2}({\Bbb{R}})\)

*with*

*Then*,

*we have the following*.

- (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

*a*be a sequence given by

(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].

*Decay*. Theorems 3.3-3.4 can be applied to many mild decay wavelets. For example, Daubechies’ wavelets [22], Lemarié-Meyer wavelets [23, 24], and spline wavelets of high order (orthogonal [25, 26], semi-orthogoanl [19, 27], biorthogonal [7]). In addition, we would like to emphasize an important fact. Lemma 3.1 guarantees that

*φ*is a radial decreasing \(L^{1}\)-majorant of

*ψ*[18], p.218, Chapter 5. Indeed, the boundedness of

*φ*follows from

*φ*contrasts with typical majorants of wavelets (polynomial \(C(1+x)^{-\alpha}\), \(\alpha>1\), exponential \(De^{-\beta x}\), \(\beta>0\)) by stressing that the majorant

*φ*is generated by wavelets themselves, not borrowed from specific \(L^{1}\) functions. Wojtaszczyk reported that orthogonal wavelet bases are unconditional bases for Lebesgue spaces by providing some assumptions [12]. His assumptions seem verbose in comparison to condition \(\mathcal{M}\).

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.

## Notes

### 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.

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