关于规范紧框架超小波若干问题的研究

Study on Several Problems of Normalized Tight Frame Super-wavelets

给出了规范紧框架超小波的3个重要结论。①空间L2(R)(m)中长度为m的规范紧框架超小波可扩展为较大空间L2(R)(m+1)中长度为m+1的超小波,也即空间L2(R)(m)的规范紧超小波框架可扩展为较大空间L2(R)(m+1)的规范正交基。从而为规范紧框架超小波与超小波搭起了桥梁。②借助于酉等价性,讨论了规范紧框架超小波的等价性.给出一个长度为m的规范紧框架超小波当最后一个分量用酉等价的规范紧框架小波代替时,所形成的仍然是一个规范紧框架超小波。③改进了规范紧框架超小波定义的条件,利用空间理论,给出了判定规范紧框架超小波的一个充分条件。该条件较定义中的条件相对简单,利用泛函分析对这些结论给出了证明,这些结论给处理超空间中的信号提供了重要的理论依据。

Three impotant results are given on normalized tight frame super-wavelets in the article.Firstly,A normalized tight frame super-wavelets of length in the space can be extended to a super-wavelets of length in the bigger space.That is a normalized tight super-wavelets frame in the space is extendable to an orthonormal basis in the bigger space.So the relationship between parseval super-wavelets and super-wavelets is found.Secondly,the equivalence of the normalized tight frame super-wavelets are studied by means of the unitary equivalence.The m-tuple remainly forms a normalized tight frame super-wavelets,when the final component of normalized tight frame super-wavelets is unitarily replaced by a normalized tight frame wavelet.Finally,the conditions of the normalized tight frame super-wavelets are improved.A sufficent condition on a normalized tight frame super-wavelets is given by using space theory.The three results are proofed by using functional analysis.These results function as the important theory basis for the signal processing in the space.