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On Frame Wavelet Sets and Some Related Topics 数理解析研究所講究録 1455 巻 2005 年 106-119 108 On Frame Wavelet Sets and Some Related Topics Xingde Dai and Yuanan Diao ABSTRACT. A special tyPe of frame wavelets in $L^{2}(\mathbb{R})$ or $L^{2}(\mathbb{R}^{d})$ consists of those whose Fourier transforms are defined by set theoretic functions The corresponding sets involved are called frame wavelet sets. The seemingly simple structure of the frame wavelets induced by frame wavelet sets turns to be quite rich and complicated. The study of such frame wavelets can serve as a platform for the study of more general frame wavelets In this PaPer, we review many recent results on frame wavelet sets and results obtained through the use of frame wavelet sets. We also pose some open questions in these related topics. 1. Introduction The concept of frames first appeared in the late 40’s and early $50\mathrm{J}\mathrm{s}|1\lfloor 9,29,30$]. The concept of frame wavelets is a simple combination of the concepts of wavelets and frames. Naturally, frame wavelets share a close relation with wavelets and the development and study of wavelet theory during the last two decades also brought much attention and interest to the study of frame wavelets. For recent developm ent and work on frames and some related topics, see [1, 16, 17, 18, 23, 24, 28]. In general, we can define a frame on any given separable Hilbert space. Let $H$ be such $\prime \mathcal{H}$ $\mathrm{t}^{\mathrm{r}}x_{j}$ $\in J$ $?t$ a space, then a family of elements : $J$ } in is called a frame for if there exist constants $A$ and $B$ , $0<A\leq B<\infty$ , such that for each $f\in \mathcal{H}$ we have . (1.1) $A||f||^{2} \leq\sum_{j\in J}|\langle f, x_{j}\rangle|^{2}\leq B||f||^{2}$ $A$ is called a lower frame bound and $B$ is called an upper frame bound of the frame in this case. The supremum of all lower frame bounds and the infimurn of all upper fram $\mathrm{e}$ bounds are called the optimal frame $bo\uparrow\lambda 7|_{\mathit{1}}d_{\iota}\mathrm{q}$ of the frame and will be denoted by $A_{0}$ and $B_{0}$ respectively throughout this paper. A frame is said to bc a tight frame when $A_{0}=B_{0}$ and a normalized tight frame when $A_{0}=B_{0}=1$ . By definition, any orthonormal basis in a Hilbert space is a normalized tight frame, but in general a normalized tight frame may not be an orthonormal basis. Following [14], we may define a frame wavelet under a general setting. First, a unitary system is a set of unitary operators & acting on a Hilbert space 7{ which contains the identity operator I of $B(H)$ . An element of $H$ is called a fram $\iota e$ wavelet (with respect to The first author thanks Shinzo Kawamura for inviting him and providing him partial travel support to the conferencc He also thanks Kichi-Suke Saito for organizing this good conference. 107 XINGDE DAI AND YU AN DIAO the system &) if the family {Ux : U\in &} is a frame for W. One can similarly define tight frame wavelets and normalized tight fvarne wavelets. However, for our purposes in this paper, we usually limit us to $?\{=L^{2}(\mathbb{R})$ or $\prime \mathcal{H}=L^{2}(\mathbb{R}^{d})$ . Depending on how the unitary system $\mathcal{U}$ is chosen, we may then obtain different frame wavelets. In this paper, we are mainly interested in a special tyPe of frame wavelets, namely those whose Fourier transforms are set theoretic functions. More precisely, let $E$ be a $L^{2}(\mathbb{R})$ the specified unitary system $\mathcal{U}$ ), then the set $E$ is called a $f\dot{r}ame$ wavelet set (for ). Similarly, $E$ is called a (normalized) tight frame wavelet set if $\psi$ is a (normalized) tight frame wavelet. The case of $L^{2}(\mathbb{R}^{d})$ can bc similarly treated. 2. Frame Wavelets with Respect to Dilation and Translation Operators We define the dilation and translation operators, $D$ and $T$ on $L^{2}(\mathbb{R})$ as follows. (Df) $(_{\backslash }x)=\sqrt{2}f(2x)$ (Tf)(x) $=f(x-1)$ , . $\mathrm{i}.\mathrm{e}_{)}.||Df||=||Tf||=||f||$ $\in L^{2}(\mathbb{R})$ $D$ $T$ for any for any $f$ . , are both unitary operators, $\{D^{n}T^{\ell} : n, l \in \mathbb{Z}\}=\mathcal{U}(D, T)$ system. A frame $f\in L^{2}(\mathbb{R})$ . Thus defines a unitary $\psi\in L^{2}(\mathbb{R})$ $\mathcal{U}(d, t)$ wavelet for $\mathcal{H}=L^{2}(_{\backslash }\mathbb{R})$ with respect to is simply a function such that (2.1) $\{D^{n}T^{p}\psi : n, l \in \mathbb{Z}\}=\{2^{\frac{n}{\underline{9}}}\psi(2^{n}x-\ell) : n, \ell\in \mathbb{Z}\}$ $0<A\leq B<\infty$ is a frame of $L^{2}(\mathbb{R})$ . In other word, there exist such that $D^{n}T^{\ell}\psi\}|^{2}\leq B||f||^{2}$ (2.2) $A||f \cdot||^{2}\leq\sum_{n,\ell\in \mathrm{Z}}|\langle f,$ $L^{2}(\mathbb{R})$ usually refers to a for all $f\in L^{2}(\mathbb{R})$ In the literature, the term frame wavelet for frame wavelet under this setting when the unitary system $\mathcal{U}$ is not specified 2.1. The Characterization Problem. Similar to the study of wavelets, an essential question in the study of frame wavelets is how to characterize them under the given unitary operating system. Under the setting of this section, the question is how to characterize a $\ell\in \mathbb{Z}\}$ $\mathcal{U}=\{D^{n}T^{\ell}$: n, . This function $\psi\in L^{2}(\mathbb{R})$ that is a frame wavelet with respect to problem remains open. However, a characterization of a normalized tight frame wavelet has been obtained [ $23_{\mathrm{J}}^{\rceil}$ and we state the result below. $C_{\psi}(\mathcal{U}(D, T))$ at THEOREM 2.1. Let $\psi$ be a fixed wavelet and let be the local commutant $T$ $L^{2}(\mathbb{R})$ $C_{\psi}(\mathcal{U}(D,T))$ operators acting on such $\psi$ , that is, is the set of all bounded linear $f\in L^{2}(\mathbb{R})$ $norm_{!}al,\mathrm{i}ze_{J}d$ tight that UT-TU $=0$ for any $U\in \mathcal{U}(D, T)$ . Then is a frame $A^{*}$ $A\in C_{\psi}(\mathcal{U}(D,T))(\dot{\iota}.e.)$ is an isomeiry) wavelet if and only if there is a $co$ -isometry such that $f=A\psi$ . Furthermore, the following theorem provides a sufficient condition for a frame wavelet. 108 ON FRAME WAVELET SETS AND SOME RELATED TOPICS THEOREM 2.2. [24] For $\psi\in L^{2}(\mathbb{R})_{f}$ define $\underline{S}_{\psi}=\mathrm{e}\mathrm{s}\mathrm{s}\inf_{\xi\in 1\mathrm{R}}\sum_{\mathrm{i}\in \mathrm{Z}}|\hat{\psi}(2^{\mathrm{j}}\xi)|^{2}$ , $\overline{\mathrm{S}}_{\psi}=\mathrm{e}\mathrm{s}\mathrm{s}\sup_{\xi\in \mathrm{I}\mathrm{R}}\sum_{\mathrm{j}\in \mathrm{Z}}|\hat{\psi}(2^{i}\xi)|^{2}$ and $\beta_{\psi}(m)=\mathrm{e}\mathrm{s}\mathrm{s}\sup_{\xi\in \mathbb{R}}\sum_{\mathrm{k}\in \mathrm{Z}}|\sum_{\mathrm{j}=0}^{\infty}\hat{\psi}(2^{\mathrm{j}}2^{\mathrm{k}}\xi)\hat{\psi}(2^{\mathrm{j}}(2^{\mathrm{k}}\xi+2\mathrm{m}\pi))|$ . If $A_{\psi}= \underline{S}_{\psi}-\sum_{q\in 2\mathrm{Z}+1}[\beta_{\psi}(q)\beta_{\psi}(-q)]\frac{1}{2}>0$ , and $B_{\psi}= \overline{S}_{\psi}+\sum_{q\in 2\mathrm{Z}+1}[\beta_{\psi}(q)\beta_{\psi}(-q)]^{\frac{1}{2}}<\infty$ , then $\psi$ is a frame wavelet with $A_{\psi}$ as a lower frame bound and $B_{\psi}$ as an upper frame bound. Note that $A_{\psi}$ and $B_{\psi}$ in the above theorem are not necessarily the optimal bounds in general. 2.2. Frame Wavelet Sets with Respect to Dilation and Translation Opera- tors. In an attempt to better understand the frame wavelets, we then turn our attention to study a much simpler subclass of frame wavelets whose Fourier transforms are simply set theoretic functions. More specifically, let $E$ be a Lebesgue measurable set of finite $\psi\wedge$ measure. Define $\psi\in L^{2}(\mathbb{R})$ by $\hat{\psi}=\frac{1}{\sqrt{2\pi}}\chi_{E)}$ where is the Fourier transform of $\psi$ . If $\psi$ so defined is a frame wavelet for $L^{2}(\mathbb{R})$ (with respect to the dilation and translation operators), then the set $E$ is called a frame wavelet set (for $L^{2}(\mathbb{R})$ ). Similarly, $E$ is called a (normalized) tight frame wavelet set if $\psi$ is a (normalized) tight frame wavelet, We wish to characterize (tight, normalized tight) frame wavelet sets. Toward this direction, a characterization of the tight frame wavelet sets has been successfully obtained (this would include the normalized tight frame wavelet sets as a special case) , and some fairly useful necessary conditions and sufficient conditions for frame wavelet sets arc also obtained. However, the characterization of frame wavelet sets in general remains an open question. Before we state these results, wc will need to introduce some terms and definitions. Let $\mathcal{F}$ be the Fourier-Plancherel transform on $\prime H$ $=L^{2}(\mathbb{R})$ : if $f$ , $g\in L^{1}(\mathbb{R})$ $\cap L^{2}(\mathbb{R})$ , then $( \mathcal{F}f)(s)=\frac{1}{\sqrt{2\pi}}\mathit{1}_{\mathrm{R}}^{e^{-ist}f(t)d\mathrm{h}}$ $=\hat{f}(s)$ , and $(F^{-1}g)(t)= \frac{1}{\sqrt{2\pi}}f_{\mathrm{R}}e^{ist}g(s)ds$ .
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