Band-Limited Wavelets with Subexponential Decay
Band-Limited Wavelets with Subexponential Decay
复制标题
DOI：
10.4153/cmb-1998-053-8
复制
发表时间：
1998-12
期刊：
影响因子：
--
通讯作者：
Jacek Dziubański;E. Hernández
中科院分区：
文献类型：
--
作者：
Jacek Dziubański;E. Hernández
摘要 众所周知，紧支持小波不能属于${{C}^{\infty }}\,(\text{R})\,\cap \,{{L}^{2}}\,(R)$类。对于具有指数衰减的小波也是如此。我们证明可以构造 ${{C}^{\infty }}\,(\text{R})\,\cap \,{{L}^{2}}\,(R)$ 类中的小波，它们“几乎”呈指数衰减，而且它们是带限的。我们通过证明我们可以调整在 $[\text{BSW}]$ 中找到的 Lemarié-Meyer 小波 $[\text{LM }\!\!]\!\!\text{ }$ 的构造，以便我们获得 $R$ 上具有次指数衰减的带限 ${{C}^{\infty }}$ 小波，即对于每个 $0\,0$ 使得 $|\psi (x)|\le {{C}_{\varepsilon }}{{e}^{-|x{{|}^{1-\varepsilon }}}}$ 、 $x\in \text{R}$ 。此外，它的所有导数也具有次指数衰减。该证明是建设性的并使用 Gevrey 函数类。
Abstract It is well known that the compactly supported wavelets cannot belong to the class ${{C}^{\infty }}\,(\text{R})\,\cap \,{{L}^{2}}\,(R)$ . This is also true for wavelets with exponential decay. We show that one can construct wavelets in the class ${{C}^{\infty }}\,(\text{R})\,\cap \,{{L}^{2}}\,(R)$ that are “almost” of exponential decay and, moreover, they are band-limited. We do this by showing that we can adapt the construction of the Lemarié-Meyer wavelets $[\text{LM }\!\!]\!\!\text{ }$ that is found in $[\text{BSW}]$ so that we obtain band-limited, ${{C}^{\infty }}$ -wavelets on $R$ that have subexponential decay, that is, for every $0\,0$ such that $|\psi (x)|\le {{C}_{\varepsilon }}{{e}^{-|x{{|}^{1-\varepsilon }}}}$ , $x\in \text{R}$ . Moreover, all of its derivatives have also subexponential decay. The proof is constructive and uses the Gevrey classes of functions.