Hilbert 空间L2(R)上的小波保持子

J4 ›› 2012, Vol. 47 ›› Issue (4): 57-61.

Hilbert 空间L2(R)上的小波保持子

银俊成^1,2,曹怀信¹

1.陕西师范大学数学与信息科学学院，陕西西安 710062；
2.中国计量学院理学院，浙江杭州 310018

收稿日期:2011-07-24 出版日期:2012-04-20 发布日期:2012-06-28
作者简介:银俊成(1975- ),男，博士研究生，讲师，研究方向为小波分析、量子计算. Email: yjccjlu@gmail.com
基金资助:
国家自然科学基金资助项目(10571113；10871224)

Wavelet preservers on the Hilbert space L²(R)

YIN Jun-cheng^1,2, CAO Huai-xin¹

1. College of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710062, Shaanxi, China;
2. College of Science, China Jiliang University, Hangzhou 310018, Zhejiang, China

Received:2011-07-24 Online:2012-04-20 Published:2012-06-28

摘要/Abstract

摘要：

研究了小波的运算性质与保持小波的算子的性质。首先, 研究了函数空间L2(R)中的全体小波构成的集合W(L2(R))的代数性质, 证明了GW(L²(R)):=W(L2(R))∪｛0｝(0与小波之集)在数乘、加法及卷积运算下是封闭的, 进而形成一个交换赋范代数; 其次, 讨论了Hilbert空间L2(R)上将小波映射为小波的有界线性算子(称为小波保持子)的性质, 证明了这些算子的全体WP(L²(R))构成一个含幺乘法半群; 最后,研究了L²(R)上将小波映射为小波或0函数的有界线性算子(称为广义小波保持子),证明了这些算子的全体GWP(L²(R))构成了Banach算子代数B(L²(R))的一个含幺赋范子代数。同时,还给出了L²(R)上有界算线性算子成为小波保持子的一个充分条件。

关键词: 小波; 小波保持子; 半群; 赋范代数

Abstract:

The operational properties of mother wavelets in the Hilbert space L²(R) and operators that preserve mother wavelets are discussed. Denoted by W(L²(R)) the set of all mother wavelets in L²(R), it is proved that GW(L²(R)):=W(L²(R))∪｛0｝ becomes an abelian normed algebra with the usual addition, scalar multiplication, convolution and the norm ‖·‖1. A bounded linear operator A on L2(R) is said to be a wavelet preserver if it maps W(L²(R)) into W(L²(R)). It is proved that the set WP(L²(R)) of all wavelet preservers is a unital multiplicative semigroup. Furthermore, an operator in B(L²(R)) is said to be a generalized wavelet preserver (GWP) if it maps GW(L²(R)) into GW(L²(R)). It is shown that the set GWP(L²(R)) of all GWPs is a unital normed subalgebra of the Banach algebra B(L²(R)). Finally, a sufficient condition for an operator to be a wavelet preserver is obtained.

Key words: wavelet; wavelet preserver; semigroup; normed algebra

银俊成1,2,曹怀信1. Hilbert 空间L2(R)上的小波保持子[J]. J4, 2012, 47(4): 57-61.

YIN Jun-cheng1,2, CAO Huai-xin1. Wavelet preservers on the Hilbert space L²(R)[J]. J4, 2012, 47(4): 57-61.

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