Connectivity of wavelets

Abstract

In this paper, path-connectivity of the set of some special wavelets in L^2\left(\mathbb{R}\right), which is the topological geometric property of wavelets, is introduced. In particular, the main progress of wavelet connectivity in the past twenty years is reviewed and some unsolved problems are listed. The corresponding results of high dimension case and other cases are also briefly explained.

Keywords

Wavelet / MRA wavelet / S–elementary wavelet / frame wavelet / path-connectivity

1 Introduction

The path-connectivity of the set composed of some special wavelets as a subset of L^2(\mathbb{R}) is one of the important research topics of wavelet theory in L^2(\mathbb{R}). International famous mathematician Weiss’s research team [20, 21] and Dai and Larson [12] independently put forward the research content for the first time and emphasized its research significance. Subsequently, the research team led by Weiss and some members of the research team led by Dai and Larson (including visiting scholars and their students) joined the research on this topic. In 1998, the two teams jointly published the results obtained in the name of “The Wutam Consortium” for the first time [39]: the set composed of all MRA wavelets is a path-connected subset in L^2(\mathbb{R}) (Theorem 1.1). The research methods of the two teams are different, the former focuses on the method of Fourier analysis, while the latter focuses on the means of functional analysis. In the past two decades, many advances have been made in this field, but many even the most basic problems remain unsolved. This paper will review the main progress of this research, list some unsolved problems and briefly explain the corresponding results of the high dimension case and other cases.

Definition 1.1　Suppose that \psi \in L^2(\mathbb{R}) and {\psi }_{j,k}(x)=2^{\frac{j}{2}}\psi (2^{j}x-k), where \mathbb{R} denotes the real number set, j,k\in \mathbb{Z} and \mathbb{Z} is the integer set. If there are constants \alpha >0 and such that

(1.1)\alpha \Vert f|{|}^2⩽\sum _{j,k\in \mathbb{Z}}|⟨f,{\psi }_{j,k}⟩{|}^2⩽\beta \Vert f{\Vert }^2,\mathrm{\forall }f\in L^2(\mathbb{R}).

Then the function family \{{\psi }_{j,k}(x)|j,k\in \mathbb{Z}\} is called a wavelet frame basis for L^2(\mathbb{R}) with bounds \alpha and \beta, abbreviated as the wavelet frame. Accordingly, the function \psi is called a wavelet frame generation function, which is called a frame wavelet for short.

If only the right inequality holds in (1.1), then \{{\psi }_{j,k}(x)|j,k\in \mathbb{Z}\} is called a Bessel sequence for L^2(\mathbb{R}) with bound \beta, and \psi is called the generator of this Bessel sequence. If \alpha =\beta in (1.1), then \{{\psi }_{j,k}(x)|j,k\in \mathbb{Z}\} is called a tight wavelet frame, and \psi is called a tight frame wavelet with constant \alpha. In particular, If \alpha =\beta =1 in (1.1), then \{{\psi }_{j,k}(x)|j,k\in \mathbb{Z}\} is called a Parseval wavelet frame, and \psi is called a Parseval frame wavelet. If \{{\psi }_{j,k}(x)|j,k\in \mathbb{Z}\} is a Riesz basis (an orthonormal basis) for L^2(\mathbb{R}), then the function \psi is called a Riesz wavelet (wavelet).

Over the past decade, the research on wavelet, wavelet frame and their applications has been very active in the fields of applied mathematics, computational mathematics and information processing, and the results are also quite rich. See [5, 18, 24, 27].

Set

　　 W^{\mathrm{B}}=\{\psi |\psi \mathrm{i}\mathrm{s}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{o}\mathrm{f}\mathrm{a}\mathrm{B}\mathrm{e}\mathrm{s}\mathrm{s}\mathrm{e}\mathrm{l}\mathrm{s}\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e}\},

　　 W^{\mathrm{F}}=\{\psi |\psi \mathrm{i}\mathrm{s}\mathrm{a}\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{m}\mathrm{e}\mathrm{w}\mathrm{a}\mathrm{v}\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{t}\},

　　 W^{\mathrm{T}\mathrm{F}}=\{\psi |\psi \mathrm{i}\mathrm{s}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{m}\mathrm{e}\mathrm{w}\mathrm{a}\mathrm{v}\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{t}\},

　　 W^{\mathrm{P}\mathrm{F}}=\{\psi |\psi \mathrm{i}\mathrm{s}\mathrm{a}\mathrm{P}\mathrm{a}\mathrm{r}\mathrm{s}\mathrm{e}\mathrm{v}\mathrm{a}\mathrm{l}\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{m}\mathrm{e}\mathrm{w}\mathrm{a}\mathrm{v}\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{t}\},

　　 W^{\mathrm{R}\mathrm{B}}=\{\psi |\psi \mathrm{i}\mathrm{s}\mathrm{a}\mathrm{R}\mathrm{i}\mathrm{e}\mathrm{s}\mathrm{z}\mathrm{w}\mathrm{a}\mathrm{v}\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{t}\},

　　 W^{\mathrm{O}\mathrm{N}\mathrm{B}}=\{\psi |\psi \mathrm{i}\mathrm{s}\mathrm{a}\mathrm{w}\mathrm{a}\mathrm{v}\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{t}\}.

Then W^{\mathrm{O}\mathrm{N}\mathrm{B}} is a subset of the unit sphere in L^2(\mathbb{R}), W^{\mathrm{P}\mathrm{F}} is a subset of the unit ball in L^2(\mathbb{R}) and the 6 sets have the following inclusion relationships [12]:

　　 W^{\mathrm{N}\mathrm{B}}⫋W^{\mathrm{R}\mathrm{B}}⫋W^{\mathrm{F}}⫋W^{\mathrm{B}},

　　 W^{\mathrm{N}\mathrm{B}}⫋W^{\mathrm{P}\mathrm{F}}⫋W^{\mathrm{T}\mathrm{F}}⫋W^{\mathrm{F}}⫋W^{\mathrm{B}}.

The wavelet connectivity problem considered in this paper refers to the path-connectivity of the six sets and their special subsets as L^2(\mathbb{R}) subsets, where the path-connectivity of the subset is defined by the following definition.

Definition 1.2　Given a subset \mathbb{S} in L^2(\mathbb{R}). If \mathrm{\forall }f,g\in \mathbb{S}, there exists \gamma (t):

\gamma :[0,1]⟶\mathbb{S},

such that \gamma (t) is continuous in L^2(\mathbb{R})-norm sense and \gamma (0)=f, \gamma (1)=g, then \mathbb{S} is called a subset of path-connectivity in topology sense of L^2(\mathbb{R})-norm.

2 Path-connectivity of wavelets

We begin to discuss the set W^{\mathrm{O}\mathrm{N}\mathrm{B}} firstly.

Problem 2.1　Is W^{\mathrm{O}\mathrm{N}\mathrm{B}} a path-connected subset in L^2(\mathbb{R})?

The problem has not been solved yet, but it has been confirmed that W^{\mathrm{O}\mathrm{N}\mathrm{B}} has two subsets that are path-connected: the set of MRA wavelets [39] and the set of S-elementary wavelets [38].

The MRA wavelet is introduced below. MRA (multiresolution analysis) is a general method for constructing wavelets. “Almost all” wavelets can be constructed by multiresolution analysis. “A few” wavelets do not come from MRA. Therefore, multirelosution analysis is one of the milestone achievements in the history of wavelet theory. Another landmark achievement is the Daubechies wavelet constructed by multiresolution analysis (See [13, 24]).

Defintion 2.1　A sequence \{V_j{\}}_{j\in \mathbb{Z}} of closed subspaces of L^2(\mathbb{R}) is called a multiresolution analysis (MRA) for L^2(\mathbb{R}) if

(1) V_j\subset V_{j+1},j\in \mathbb{Z}; f\in V_j if and only if f(2\cdot )\in V_{j+1},j\in \mathbb{Z};

(2) \bigcap _{j\in \mathbb{Z}}{V}_j=\{0\}; \overline{\bigcup _{j\in \mathbb{Z}}{V}_j}=L^2(\mathbb{R});

(3) there exists \phi \in V_0 such that \{\phi (\cdot -k)|k\in \mathbb{Z}\} is an orthonormal basis for V_0, where \phi is called a scaling function for this MRA. We write the MRA as {\left\{V_j,\phi \right\}}_{j\in \mathbb{Z}}.

For the MRA {\left\{V_j,\phi \right\}}_{j\in \mathbb{Z}}, it is affirmative that there is a funciton \psi \in V_1\ominus V_0 such that \psi is a wavelet for L^2(\mathbb{R}) [13, 24], and \psi is called an MRA wavelet. Let the set of all MRA wavelets be W_{\mathrm{M}}^{\mathrm{O}\mathrm{N}\mathrm{B}}.

Theorem 2.1 [39]　 W_{\mathrm{M}}^{\mathrm{O}\mathrm{N}\mathrm{B}} is a path-connected subset in L^2(\mathbb{R}).

The proof of Theorem 2.1 mainly depends on the characterization of wavelet and wavelet multiplier.

Now let's turn to S-elementary wavelet.

Defintion 2.2　If \psi is a wavelet for L^2(\mathbb{R}) and |\hat{\psi }|={\chi }_{\mathbb{E}}, then \psi is called an S-elementary wavelet, where \hat{\psi } is the Fourier transform of \psi, \mathbb{E} is a measurable subset in \mathbb{R} and {\chi }_{\mathbb{E}} denotes the characteristic function of \mathbb{E}. Accordingly, \mathbb{E} is called a wavelet set. Similarly, frame wavelet set, tight frame wavelet set and Parseval frame wavelet set can be defined.

Because the S-elementary wavelet has the smallest frequency support, it is also called the minimum support frequency (MSF) wavelet in [20, 21]. Let the set of all S-elementary wavelets be W_{\mathrm{S}}^{\mathrm{O}\mathrm{N}\mathrm{B}}.

Theorem 2.2 [38]　 W_{\mathrm{S}}^{\mathrm{O}\mathrm{N}\mathrm{B}} is a path-connected subset in L^2(\mathbb{R}).

The proof of Theorem 2.2 strongly depends on the characterization of a wavelet set.

It was shown in [12] that W_{\mathrm{S}}^{\mathrm{O}\mathrm{N}\mathrm{B}} is not a subset of W_{\mathrm{M}}^{\mathrm{O}\mathrm{N}\mathrm{B}} and

W_{\mathrm{M}}^{\mathrm{O}\mathrm{N}\mathrm{B}}\bigcap W_{\mathrm{S}}^{\mathrm{O}\mathrm{N}\mathrm{B}}\ne \mathrm{\varnothing }.

Problem 2.2　Is W_{\mathrm{M}}^{\mathrm{O}\mathrm{N}\mathrm{B}}\bigcap W_{\mathrm{S}}^{\mathrm{O}\mathrm{N}\mathrm{B}} a path-connected subset in L^2(\mathbb{R})?

Problem 2.2 has not been solved yet, but it has been confirmed that W_{\mathrm{M}}^{\mathrm{O}\mathrm{N}\mathrm{B}}\bigcap W_{\mathrm{S}}^{\mathrm{O}\mathrm{N}\mathrm{B}} has a path-connected subset [37].

For the Riesz wavelet, the following question was proposed by Larson.

Problem 2.3 [23]　Is W^{\mathrm{R}\mathrm{B}} a path-connected subset in L^2(\mathbb{R})?

Problem 2.3 has not been solved yet, but the results in [4] are relevant to this problem.

3 Path-connectivity of frame wavelets

The path-connectivity problem of the set W^{\mathrm{F}} has been completely solved.

Theorem 3.1 [3, 19]　 W^{\mathrm{F}} is a path-connected subset in L^2(\mathbb{R}).

The proofs of Theorem 3.1 were established by different methods in [3] and [19], respectively. The former fully uses Fourier analysis technique, while the latter uses the orthogonality of frame, Bessel sequence and the properties of wavelet Bessel multiplier.

For the tight frame wavelet set W^{\mathrm{T}\mathrm{F}}, we propose the following questions.

Problem 3.1　Is W^{\mathrm{T}\mathrm{F}} a path-connected subset in L^2(\mathbb{R})?

For Problem 3.1, we have not referred to the relevant literatures.

Similarly, for the Parseval wavelet frame set W^{\mathrm{P}\mathrm{F}}, we have the following problem.

Problem 3.2　Is W^{\mathrm{P}\mathrm{F}} a path-connected subset in L^2(\mathbb{R})?

Although Problem 3.2 has not been solved yet, it has been proved that at least four subsets of W^{\mathrm{P}\mathrm{F}} are path-connected: the three sets composed of the Parseval frame wavelets satisfying certain conditions under Fourier transform and S-elementary Parseval frame wavelet set.

This section only describes the path-connectivity of three sets composed of the Parseval frame wavelets satisfying certain conditions under Fourier transform, and the path-connectivity of the S-elementary Parseval frame wavelet set will be introduced in the next section.

First of all, assume that \psi \in L^2(\mathbb{R}), and the following two conditions are given:

(1) there exists ϵ\in (0,\pi ] such that

(3.1)

here \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\hat{\psi }=\{\xi |\hat{\psi }(\xi )\ne 0\} and |E| is the Lebesgue measure of the set E;

(2) there exists ϵ\in (0,\pi ] such that

(3.2)\left|\underset{n\to \mathrm{\infty }}{lim sup}\frac{1}{2^n}((\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\hat{\psi })\bigcap 2^n{J}_0^{ϵ})\right|=0,

where J_0^{ϵ}=(-ϵ,-\frac{ϵ}{2}]\bigcup [\frac{ϵ}{2},ϵ).

Based on conditions (3.1) and (3.2), we have the following results.

Theorem 3.2 [16]　(1) {\mathcal{K}}_{\tau }=\{\psi |\psi isaParsevalframewaveletsatisfing(3.1)\} is a path-connected subset in L^2(\mathbb{R}).

(2) {\mathcal{K}}_d=\{\psi |\psi isaParsevalframewaveletsatisfing(3.2)\} is a path-connected subset in L^2(\mathbb{R}).

It is easy to see that conditions (3.1) and (3.2) are the size conditions imposed on Fourier transform of the Parseval frame wavelet, where condition (3.1) involves the shifts of 2\pi integer multiple of the interval with the length of 2ϵ, and condition (3.2) involves the dilations of 2^n multiple of the interval with the length of 2ϵ. The necessary and sufficient conditions of Parseval frame wavelet set and analysis skills were used for many times in the proof of Theorem 3.2. In addition, if \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\hat{\psi } of the Parseval frame wavelet \psi is bounded, then \psi \in {\mathcal{K}}_{\tau }\bigcap {\mathcal{K}}_d. This shows that {\mathcal{K}}_{\tau } and {\mathcal{K}}_d include “more considerable” Parseval frame wavelets.

As an extension of the result in [16], it obtained in [15] that a larger set of the Parseval frame wavelets containing {\mathcal{K}}_{\tau } and {\mathcal{K}}_d is path-connected.

In order to present the results in [15], we need to make some preparations.

Definition 3.1　Suppose that J\subset \mathbb{R}. If \{2^{j}J{\}}_{j\in \mathbb{Z}} is a partition of \mathbb{R}\setminus \{0\} (a.e.), then J is called a Calderón set.

Example 3.1 (An example of Calderón set)　Shannon set S=[-2\pi ,-\pi )\bigcup (\pi ,2\pi ].

Assume that J,K\subset \mathbb{R}. Let

[J]=\bigcup _{j\in \mathbb{Z}}{2}^{-j}J,[J{]}_{-}=\bigcup _{j⩾0}{2}^{-j}J,{\tau }_K(J)=\bigcup _{k\in \mathbb{Z}\setminus \{0\}}(J+2k\pi )\cap K.

Definition 3.2　Suppose that J_0 is a bounded Calderón set, J_j=2^{-j}{J}_0 and K\subset \mathbb{R} is measurable. If there is an M\in \mathbb{Z} such that correspondence between sets

I\subset [J_M{]}_{-}=\bigcup _{l⩾M}{2}^{-l}{J}_0⟼\mathcal{L}(I)={\mathcal{L}}_K(I)=[{\tau }_K(I)]\cap S

is continuous in the sense of measure algebra, that is, \mathrm{\forall }\eta >0, there exists \delta >0 such that for every I\subset [J_M{]}_{-} with , then K is said to belong to the set \mathbb{M}(J_0).

The corresponding result is as follows.

Theorem 3.3 [15]　 Suppose that J_0\subset [-\pi ,\pi ] is a Calderón set. Then

\mathcal{K}(J_0)=\{\psi |\psi isaParsevalframewaveletand\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\hat{\psi }\in \mathbb{M}(J_0)\}

is a path-connected subset in L^2(\mathbb{R}).

At the same time, it was shown in [15] that {\mathcal{K}}_{\tau }\bigcup {\mathcal{K}}_d⫋\mathcal{K}(S), \mathbb{R}\notin \mathbb{M}(J_0), where J_0 is a bounded Calderón set. \mathbb{R}\notin \mathbb{M}(J_0) implies that \mathcal{K}(J_0)⫋W^{\mathrm{P}\mathrm{F}}.

The previous Section 2 introduced the MRA wavelet, and naturally it comes to mind: is there an MRA Parseval frame wavelet? First, the definition of FMRA (frame multiresolution analysis) is listed.

Definition 3.3 [2]　A sequence \{V_j{\}}_{j\in \mathbb{Z}} of closed subspaces of L^2(\mathbb{R}) is called an FMRA for L^2(\mathbb{R}) if

(1) V_j\subset V_{j+1},j\in \mathbb{Z}; f\in V_j if and only if f(2\cdot )\in V_{j+1},j\in \mathbb{Z};

(2) {\bigcap }_{j\in \mathbb{Z}}{V}_j=\{0\}; \overline{{\bigcup }_{j\in \mathbb{Z}}{V}_j}=L^2(\mathbb{R});

(3) there exists \phi \in V_0 such that \{\phi (\cdot -k)|k\in \mathbb{Z}\} is a frame for V_0, where \phi is called a scaling function for this FMRA. We write the FMRA as {\left\{V_j,\phi \right\}}_{j\in \mathbb{Z}}.

If (3) becomes the following condition:

(3′) there exists \phi \in V_0 such that \{\phi (\cdot -k)|k\in \mathbb{Z}\} is a Parseval frame for V_0,

then this FMRA is called a Parseval FMRA.

Obviously, an MRA {\left\{V_j,\phi \right\}}_{j\in \mathbb{Z}} is a Parseval FMRA, but the reverse is not true [8].

For the Parseval FMRA {\left\{V_j,\phi \right\}}_{j\in \mathbb{Z}}, there must exist a function \psi \in V_1\ominus V_0 such that \psi is a Parseval wavelet for L^2(\mathbb{R}), and \psi is called an MRA Parseval wavelet. Let W_{\mathrm{M}}^{\mathrm{P}\mathrm{F}} be the set of all MRA Parseval frame wavelets. Then we raise the following question.

Problem 3.3　Is W_{\mathrm{M}}^{\mathrm{P}\mathrm{F}} a path-connected subset in L^2(\mathbb{R})?

For Problem 3.3, we have not consulted the relevant literature. There is also another definition of the so-called MRA Parseval frame wavelet, which defined the set of MRA Parseval frame wavelets contains W_{\mathrm{M}}^{\mathrm{P}\mathrm{F}}, and this set is a path-connected subset in L^2(\mathbb{R}) [35, 36].

4 Path-connectivity of S-elementary frame wavelets

The following three sets are the main subsets of S-elementary frame wavelets.

　　 W_{\mathrm{S}}^{\mathrm{F}}=\{\psi |\psi \mathrm{i}\mathrm{s}\mathrm{a}\mathrm{n}\mathrm{S}-\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{y}\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{m}\mathrm{e}\mathrm{w}\mathrm{a}\mathrm{v}\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{t}\},

　　 W_{\mathrm{S}}^{\mathrm{T}\mathrm{F}}=\{\psi |\psi \mathrm{i}\mathrm{s}\mathrm{a}\mathrm{n}\mathrm{S}-\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{y}\mathrm{t}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{m}\mathrm{e}\mathrm{w}\mathrm{a}\mathrm{v}\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{t}\},

　　 W_{\mathrm{S}}^{\mathrm{T}\mathrm{F}}(k)=\{\psi |\psi \mathrm{i}\mathrm{s}\mathrm{a}\mathrm{n}\mathrm{S}-\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{y}\mathrm{t}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{m}\mathrm{e}\mathrm{w}\mathrm{a}\mathrm{v}\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{t}\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}k\},

where k is a positive integer. It should be noted that the constants of S-elementary frame wavelets must be positive integers [7].

It is clear to see that

W_{\mathrm{S}}^{\mathrm{T}\mathrm{F}}(k)⫋W_{\mathrm{S}}^{\mathrm{T}\mathrm{F}}⫋W_{\mathrm{S}}^{\mathrm{F}},

and if k=1, then W_{\mathrm{S}}^{\mathrm{T}\mathrm{F}}(k)=W_{\mathrm{S}}^{\mathrm{P}\mathrm{F}}, here

W_{\mathrm{S}}^{\mathrm{P}\mathrm{F}}=\{\psi |\psi \mathrm{i}\mathrm{s}\mathrm{a}\mathrm{n}\mathrm{S}-\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{y}\mathrm{P}\mathrm{a}\mathrm{r}\mathrm{s}\mathrm{e}\mathrm{v}\mathrm{a}\mathrm{l}\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{m}\mathrm{e}\mathrm{w}\mathrm{a}\mathrm{v}\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{t}\}.

Therefore, based on the feature descriptions of frame wavelet sets and tight frame wavelet sets, the following results were obtained in [9] and [6], respectively.

Theorem 4.1 [9]　 W_{\mathrm{S}}^{\mathrm{F}} is a path-connected subset in L^2(\mathbb{R}).

Theorem 4.2 [6]　 W_{\mathrm{S}}^{\mathrm{T}\mathrm{F}}(k) is a path-connected subset in L^2(\mathbb{R}), but W_{\mathrm{S}}^{\mathrm{T}\mathrm{F}} is not connected.

Similar to Problem 2.2, we have the following problem.

Problem 4.1　Is W_{\mathrm{M}}^{\mathrm{P}\mathrm{F}}\bigcap W_{\mathrm{S}}^{\mathrm{P}\mathrm{F}} a path-connected subset in L^2(\mathbb{R})?

Since every element \psi in set W_{\mathrm{M}}^{\mathrm{P}\mathrm{F}}\bigcap W_{\mathrm{S}}^{\mathrm{P}\mathrm{F}} is an MRA Parseval frame wavelet, there must be a scaling function \varphi corresponding to \psi. Under very “natural” condition imposed to the scaling function, the following theorem was obtained.

Theorem 4.3 [11]　 W_{\mathrm{M}}^{\mathrm{P}\mathrm{F}}\bigcap W_{\mathrm{S}}^{\mathrm{P}\mathrm{F}} is a path-connected subset in L^2(\mathbb{R}).

The “natural” condition here refers to the scaling function maintaining the “path”, as detailed in [11].

5 High dimension cases

Suppose that A is a d\times d real dilation matrix (matrix elements are real numbers and the modulus of the every eigenvalue of the matrix is greater than 1). For f\in L^2({\mathbb{R}}^d), The dilation operator and shift operator are defined as follows, respectively.

D_{A}f(t)=\sqrt{|\mathrm{d}\mathrm{e}\mathrm{t}A|}f(At),T_{k}f(t)=f(t-k),k\in {\mathbb{Z}}^d.

\psi in L^2({\mathbb{R}}^d) is called a frame wavelet for L^2({\mathbb{R}}^d) if there are positive constants \alpha and \beta such that

\mathrm{\forall }f\in L^2({\mathbb{R}}^d),\alpha \Vert f|{|}^2⩽\sum _{j\in \mathbb{Z},k\in {\mathbb{Z}}^d}|⟨f,D_A^j{T}_k\psi ⟩{|}^2⩽\beta \Vert f{\Vert }^2.

Terms such as high dimension Bessel sequence generators, tight frame wavelets, Parseval frame wavelets, Risez wavelets and wavelets can be similarly defined. Accordingly, we can discuss the path-connectivity problem of these wavelet classes in L^2({\mathbb{R}}^d). Due to the complexity of high dimension space geometric structure and the dilation matrix, only a part of the previously stated results can be obtained in high dimension cases.

In [31, 32], Theorem 2.1 was extended to high dimension cases, but it is necessary to assume that A is a d\times d integer dilation matrix and |\mathrm{d}\mathrm{e}\mathrm{t}A|=2. The method in [31] works for all high dimension cases.

The high dimension cases were also discussed while obtaining Theorem 2.2, Theorem 3.1, and Theorem 4.1 in [38], [3, 19], and [9], respectively.

In [11], the same result as in Theorem 4.3 was obtained for high dimension arbitrary real dilation matrices.

For a 2\times 2 integer dilation matrix A with |\mathrm{d}\mathrm{e}\mathrm{t}A|=2, A can be divided into six equivalence classes, each of which has a simple representative element. Moreover, this kind of dilation plane wavelets is more useful. So the papers [25, 26, 28, 32] reported its path-connectivity. For d>2, although a d\times d integer dilation matrix A with |\mathrm{d}\mathrm{e}\mathrm{t}A|=2 is complex, some results have been obtained (see [29, 34]).

6 Other cases

For path-connectivity of super frame wavelets and multiwavelets, see [14, 30, 33]. In addition, there also exist the following problems.

Problem 6.1　Can the path constructed by proving path-connectivity be a direct path?

This question was raised and discussed in [1]. See [1, 10] for relevant results.

Problem 6.2　Is the previously listed every subset of path-connectivity uniformly path-connected?

For the discussion of Problem 6.2, see [12, 17].

Finally, we give the results of Gabor wavelet connectivity.

Theorem 6.1 [22]　 The sets of functions that generate the Gabor frame, the Gabor Parseval frame and the Gabor orthonormal basis are all path-connected subsets in L^2(\mathbb{R}), respectively.

References

[1] Azoff E, Ionascu E, Larson D. . Direct paths of wavelets. Houston J Math 2003; 29: 737–756

[2] Benedetto J, Li S D. The theory of multiresolution analysis frames and applications to filter banks. Appl Comput Harmonic Anal 1998; 5(4): 389–427

[3] Bownik M. Connectivity and density in the set of framelets. Math Res Lett 2007; 14: 285–293

[4] Cabrelli C, Molter U. Density of the set of generators of wavelet systems. Constr Approx 2007; 26: 65–81

[5] ChristensenO. An Introduction to Frames and Riesz Bases. 2nd ed. Boston: Birkhäuser, 2016

[6] Dai X D, Diao Y N. The path-connectivity of s-elementary tight frame wavelets. J Appl Func Anal 2007; 2(4): 309–316

[7] Dai X D, Diao Y N, Gu Q. Frame wavelet sets in R. Proc Amer Math Soc 2001; 129(7): 2045–2055

[8] Dai X D, Diao Y N, Gu Q, Han D G. Wavelets with frame multiresolution analysis. J Fourier Anal Appl 2003; 9(1): 39–48

[9] Dai X D, Diao Y N, Gu Q, Han D G. The s-elementary frame wavelets are path-connected. Proc Amer Math Soc 2004; 132(9): 2567–2575

[10] Dai X D, Diao Y N, Guo X X. On the direct path problem of s-elementary frame wavelets. Sci China Math 2010; 53(12): 3187–3196

[11] Dai X D, Diao Y N, Li Z Y. The path-connectivity of s-elementary frame wavelets with frame MRA. Acta Appl Math 2009; 107: 203–210

[12] Dai X D, Larson D. Wandering vectors for unitary systems and orthogonal wavelets. Memoirs Amer Math Soc 1998; 134(640): (68 pp)

[13] DaubechiesI. Ten Lectures on Wavelets. Regional Conference Series in Applied Mathematics, Vol 61. Philadelphia, PA: SIAM, 1992

[14] Diao Y N, Li Z Y. On S-elementary super frame wavelets and their path-connectedness. Acta Appl Math 2011; 116: 157–171

[15] Garrigós G, Hernández E, Šikić H, Soria F. Further results on the connectivity of Parseval frame wavelets. Proc Amer Math Soc 2006; 134(11): 3211–3221

[16] Garrigós G, Hernández E, Šikić H., Soria F, Weiss G. Connectivity in the set of tight frame wavelets (TFW). Glas Mat 2003; 38(58): 75–98

[17] Gu Q. On interpolation families of wavelets. Proc Amer Math Soc 2000; 128(10): 2973–2979

[18] HanB. Framelets and Wavelets: Algorithms, Analysis, and Applications. Cham: Birkhäuser, 2017

[19] Han D G, Larson D. On the orthogonality of frames and the density and connectivity of wavelet frames. Acta Appl Math 2009; 107: 211–222

[20] Hernández E, Wang X, Weiss G. Smoothing minimally supported frequency wavelets I. J Fourier Anal Appl 1996; 2(4): 329–340

[21] Hernández E, Wang X, Weiss G. Smoothing minimally supported frequency wavelets II. J Fourier Anal Appl 1997; 3(1): 23–41

[22] Labate D, Wilson E. Connectivity in the set of Gabor frames. Appl Comput Harmonic Anal 2005; 18(1): 123–136

[23] LarsonD. Unitary systems and wavelets sets. In: Qian T, Vai M I, Xu Y, eds. Wavelet Analysis and Applications. Appl Numer Harmonic Anal, Basel: Birkhäuser, 2007, 143–171

[24] LiD F. Mathematical Theory of Wavelet Analysis. Beijing: Science Press, 2017 (in Chinese)

[25] Li D F, Cheng J F. Some applications of E-wavelet multipliers. Chinese Quart J Math 2004; 19(3): 292–299

[26] Li D F, Cheng J F. Construction of MRA E-tight frame wavelets, multipliers and connectivity properties. Int J Wavelets Multiresolut Inf Process 2011; 9(5): 713–729

[27] LiD FXueM Z. Bases and Frames in Banach Spaces. Beijing: Science Press, 2007 (in Chinese)

[28] Li Y Z. On a class of bidimensional nonseparable wavelet multipliers. J Math Anal Appl 2002; 270: 543–560

[29] Li Y Z, Xue Y Q. The equivalence between seven classes of wavelet multipliers and arcwise connectivity they induce. J Fourier Anal Appl 2013; 19: 1060–1077

[30] Li Z Y, Dai X D, Diao Y N. Intrinsic s-elementary Parseval frame multiwavelets in L2(Rd). J Math Anal App 2010; 367: 677–684

[31] Li Z Y, Dai X D, Diao Y N, Huang W. The path-connectivity of MRA wavelets in L2(Rd). Illinois J Math 2010; 54: 601–620

[32] Li Z Y, Dai X D, Diao Y N, Xin J G. Multipliers, phases and connectivity of MRA wavelets in L2(R2). J Fourier Anal Appl 2010; 16: 155–176

[33] Li Z Y, Shi X L. On Parseval super frame wavelets. Appl Math J Chinese Univ 2012; 27(2): 192–204

[34] Li Z Y, Shi X L. Parseval frame wavelet multipliers in L2(Rd). Chin Ann Math Ser B 2012; 33: 949–960

[35] Paluszyński M, Šikić H, Weiss G, Xiao S. Generalized low pass filters and MRA frame wavelets. J Geom Anal 2001; 11: 311–342

[36] Paluszyński M, Šikić H, Weiss G, Xiao S. Tight frame wavelets, their dimension functions, MRA tight frame wavelets and connectivity properties. Adv Comput Math 2003; 18: 297–327

[37] Singh D. Path-connectivity of two-interval MSF wavelets. Kyungpook Math J 2011; 51: 293–300

[38] Speegle D. S-elementary wavelets are path-connected. Proc Amer Math Soc 1999; 127(1): 223–233

[39] The Wutam Consortium. Basic properties of wavelets. J Fourier Anal Appl 1998; 4: 575–594

RIGHTS & PERMISSIONS

Higher Education Press 2023