Lipschitz空间Λ_α()和Zygmund函数类Λ_* ()的紧小波框架系数刻画

Lipschitz空间 Λ α ( http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459564&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459564&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459564&type=middle )和Zygmund函数类 Λ * ( http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459567&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459567&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459567&type=middle )的紧小波框架系数刻画 Characterizations of the Lipschitz Spaces Λ α ( http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459569&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459569&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459569&type=middle ) and the Zygmund Class Λ * ( http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459570&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459570&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459570&type=middle ) in Terms of Tight Wavelet Frame Coefficients 1 first-author 鲁大勇, 男, 博士, 主要从事小波分析及图像处理方面的研究. E-mail: dayonglu@163.com 鲁大勇, 男, 博士, 主要从事小波分析及图像处理方面的研究. E-mail: dayonglu@163.com dayonglu@163.com 1 2 河南大学 数学与统计学院, 河南 开封 475001 河南大学 数学与统计学院, 河南 开封 475001 School of Mathematics and Statistics, Henan University, Kaifeng 475001, Henan, China School of Mathematics and Statistics, Henan University, Kaifeng 475001, Henan, China 武汉纺织大学 数学与计算机学院, 湖北 武汉 430200 武汉纺织大学 数学与计算机学院, 湖北 武汉 430200 School of Mathematics and Computer, Wuhan Textile University, Wuhan 430200, Hubei, China School of Mathematics and Computer, Wuhan Textile University, Wuhan 430200, Hubei, China 本文通过赋予母框架小波一定的正则性, 利用卷积的知识, 证明了Lipschitz空间 Λ α ( http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459571&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459571&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459571&type=middle )和Zygmund函数类 Λ * ( http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459574&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459574&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459574&type=middle )这两个函数空间可以通过紧小波框架分解系数来刻画. By means of the knowledge of convolution and endowing the mother wavelet frames with a certain degree of regularity, the paper shows that the Lipschitz spaces Λ α ( http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459576&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459576&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459576&type=middle ), 0 < α < 1, and the Zygmund class Λ * ( http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459577&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459577&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459577&type=middle ) can be characterized in terms of tight wavelet frame coefficients. Lipschitz空间 Zygmund类 框架 紧小波框架 Lipschitz spaces Zygmund class frames tight wavelet frames 引言 1 20世纪80年代，小波的出现极大地推动了非线性逼近理论的发展.它可以把目标函数快速、方便地分解为一列函数构造块，通常称其为小波级数.利用Schmidt在1907年提出的 n -项逼近理论 [ 1 1 ] ，根据目标函数的要求选取合适的项，并且控制选取项的个数. 从函数逼近论的角度看，小波可以为许多重要的函数空间(如Lebesgue空间、Hardy空间、Sobolev空间、Besov空间、Triebel-Lizorkin空间)提供无条件基(小波基)并且可以依照小波系数刻画上述函数空间 [ 2 2 , 3 3 ] .小波的上述结果从理论上来看非常完美，但是在应用中却有着很大的不足之处.例如在图像去噪过程中，如果把图像函数视为某函数空间(如Besov空间)的一个函数而去用小波系数刻画它，所用到的小波具有很高的消失距，得到的恢复图像会产生类似于锯齿形状的边缘效应.产生上述缺陷的一个重要原因来自小波基的ω-无关性.Gribonval和Nielsen等人在2004年指出 [ 4 4 ] ：如果用小波框架替代传统的小波基，所用到的母框架小波可以有更低的消失矩，可以很好地压制边缘效应.小波框架在应用中的优势对理论研究提出了要求，即要求用小波框架系数去刻画前面所提到的函数空间. 目前，如下几个函数空间已经完成了小波框架系数的刻画：2004年Borup，Gribonval和Nielsen依照小波框架系数刻画了Lebesgue空间和Sobolev空间 [ 5 5 ] ；同年，Borup，Gribonval和Nielsen又给出了Besov空间的一个小波框架系数刻画 [ 6 6 ] ；Han和Shen在2009年给出了Sobolev空间中双框架小波的构造方案 [ 7 7 ] ，通过该方案可以很容易地把Sobolev空间中的函数展成双小波框架级数，从而可以用此类双小波框架刻画Sobolev空间中的函数；鲁大勇和樊启斌在2010年使用Littlewood-Paley函数给出了Lebesgue空间的紧小波框架系数刻画 [ 8 8 ] .许多已有的结论显示依照小波框架系数刻画函数空间与依照小波基系数刻画函数空间有一定的相似性.但是由于小波框架是冗余的，因此它们之间又是有区别的，于是，研究两者之间的关系及性质的联系和区别也成为了一件非常有意义的事. 由于紧小波框架的典则对偶框架与它本身只是相差一个常数，在函数的分解与重构过程中使用起来非常方便，因此本文将使用紧小波框架系数去刻画Lipschitz空间和Zygmund函数类.关于紧小波框架的研究目前已有大量的文献，参见文献[ 9 9 ~ 14 14 9~14 ].构造紧小波框架的方案主要有Ron和Shen提出的UEP方案 [ 10 10 ] 以及Daubichies和Han等人提出的OEP方案 [ 11 11 ] . 准备知识 1 文中 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459578&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459578&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459578&type=middle 表示实数集合； http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459579&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459579&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459579&type=middle 表示整数集合； http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459583&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459583&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459583&type=middle 表示自然数集合； l 是一个取定的正整数. 设 H 为一个可分的Hilbert空间，赋予内积和范数为〈·, ·〉和 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459587&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459587&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459587&type=middle .令 J 为一个可数指标集以及{ φ j } j ∈ J ⊂ H .如果存在常数 A 和 B ，0＜ A ≤ B ＜∞，使得对任意的 x ∈ H 有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459336&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459336&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459336&type=middle 那么称{ φ j } j ∈ J 为 H 的一个界是 A 和 B 的框架.如果 A = B ，那么称该框架为紧框架.如果 A = B =1，那么称该框架为规范紧框架.由于紧框架都可以转化为规范紧框架，因此本文主要讨论规范紧框架.通常取 H 为 L 2 ( http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459590&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459590&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459590&type=middle ). 令 Ψ ={ ψ 1 , ψ 2 , …, ψ l }⊂ L 2 ( http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459592&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459592&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459592&type=middle )，对应的小波函数系为 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459341&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459341&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459341&type=middle 其中 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459594&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459594&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459594&type=middle .如果 X ( Ψ )生成 L 2 ( http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459596&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459596&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459596&type=middle )的一个规范紧框架，则称其为规范紧小波框架并有如下结果： 定理1 [ 15 15 ]    X ( Ψ )生成 L 2 ( http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459600&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459600&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459600&type=middle )的一个规范紧小波框架当且仅当 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459346&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459346&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459346&type=middle 和 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459351&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459351&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459351&type=middle 成立. 对0＜ α ＜1，定义Lipschitz空间 Λ α ≡ Λ α ( http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459607&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459607&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459607&type=middle )为 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459356&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459356&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459356&type=middle Λ α 中的范数定义为 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459361&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459361&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459361&type=middle 可以证明( Λ α , ‖·‖ Λ α )是一个Banach空间 [ 16 16 ] . 当 α =1时，对应的空间 Λ 1 不完备，因而我们引入Zygmund函数类.所有满足 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459364&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459364&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459364&type=middle 的函数 f ∈ L ∞ ( http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459612&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459612&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459612&type=middle )所组成的集合称为Zygmund函数类，记为 Λ * = Λ * ( http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459617&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459617&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459617&type=middle ). (7)式中不等式的左边是该Zygmund函数类的范数，记为 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459619&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459619&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459619&type=middle . 设 s 为一个非负整数.称 f 属于正则类 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459624&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459624&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459624&type=middle 当且仅当存在常数 ε ＞0, γ ＞0, 0＜ C ＜∞和0＜ C n ＜∞使得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459371&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459371&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459371&type=middle 定理2   令 r ≥ ε ＞0且 n 0 ∈ http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459628&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459628&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459628&type=middle .设 g 和 h 满足 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459376&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459376&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459376&type=middle 其中 C n , 1 ，0≤ n ≤ n 0 +1和 C 2 与 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459632&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459632&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459632&type=middle 无关.则存在常数 C 使得对任意满足 j ≤ l 的 j , k , l , http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459636&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459636&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459636&type=middle 有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459381&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459381&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459381&type=middle 定理3   令 ε ＞0.设 g 和 h 满足 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459386&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459386&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459386&type=middle 和 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459391&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459391&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459391&type=middle 其中 C 1 和 C 2 与 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459640&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459640&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459640&type=middle 无关.则存在常数 C 使得对任意满足 j ≤ l 的 j , k , l , http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459644&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459644&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459644&type=middle 有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459395&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459395&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459395&type=middle 对 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459649&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459649&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459649&type=middle ，定义 f 的Fourier变换为： http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459398&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459398&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459398&type=middle 文中用*表示卷积符号，即对 f 和 g 有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459403&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459403&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459403&type=middle 需要指出的是，文中的常数根据上下文的意思是不同的. Lipschitz空间 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459654&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459654&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459654&type=middle 的刻画(0＜ α ＜1) 1 令 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459658&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459658&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459658&type=middle 满足 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459412&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459412&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459412&type=middle 定义函数空间 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459661&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459661&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459661&type=middle 为 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459414&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459414&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459414&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459664&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459664&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459664&type=middle 中的范数定义为 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459418&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459418&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459418&type=middle 其中 ψ 2 － j ( x )=2 j ψ (2 j x ), ∀ ψ ∈ Ψ . 定理4   对0＜ α ＜1有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459668&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459668&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459668&type=middle 且它们的范数等价. 证   设 f ∈ Λ α .对 ψ ∈ Ψ .因为 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459671&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459671&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459671&type=middle ，所以有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459675&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459675&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459675&type=middle 且 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459427&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459427&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459427&type=middle 再由(6)式可知 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459432&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459432&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459432&type=middle 上面 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459436&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459436&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459436&type=middle 因此 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459439&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459439&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459439&type=middle 从而 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459443&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459443&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459443&type=middle 即， http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459446&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459446&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459446&type=middle 设 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459678&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459678&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459678&type=middle .如记 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459680&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459680&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459680&type=middle ，由(12)式可得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459453&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459453&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459453&type=middle 所以 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459457&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459457&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459457&type=middle 因此有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459460&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459460&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459460&type=middle 对 ψ ∈ Ψ ， http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459463&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459463&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459463&type=middle 对 J ψ 的估计要用到条件 α ＜1.因为 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459682&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459682&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459682&type=middle ，所以 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459466&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459466&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459466&type=middle 上面最后一个不等式用到了微分中值定理.又因为 j ＜－log 2 | h |等价于2 j | h |＜1，因此有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459471&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459471&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459471&type=middle 这样就有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459475&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459475&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459475&type=middle 所以 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459478&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459478&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459478&type=middle 从而有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459481&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459481&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459481&type=middle 定理5   令0＜ α ＜1， Ψ ={ ψ 1 , ψ 2 , …, ψ l }.设 X ( Ψ )是一个紧小波框架.如果 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459684&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459684&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459684&type=middle ，那么 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459687&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459687&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459687&type=middle 当且仅当 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459691&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459691&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459691&type=middle 和 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459483&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459483&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459483&type=middle 其中 C Ψ 为与 Ψ 有关的常数. 证   由于紧小波框架都可以转化为规范紧小波框架，且二者仅相差一个常系数，所以我们在证明过程中设 X ( Ψ )是一个规范紧小波框架是合理的. 设 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459695&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459695&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459695&type=middle .对 ψ ∈ Ψ ，因为 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459488&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459488&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459488&type=middle 其中 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459699&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459699&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459699&type=middle ，所以 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459702&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459702&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459702&type=middle 有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459492&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459492&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459492&type=middle 反之，设 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459707&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459707&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459707&type=middle 满足(15)式.可以验证当 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459710&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459710&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459710&type=middle ， j ≤0和 ψ ∈ Ψ 时(15)式同样成立.事实上， http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459496&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459496&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459496&type=middle 取 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459714&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459714&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459714&type=middle .因为 X ( Ψ )是一个规范紧小波框架，所以 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459499&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459499&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459499&type=middle 因此对 j =1, 2, …, 有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459504&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459504&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459504&type=middle 由定理2可知 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459507&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459507&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459507&type=middle 因为1－ α ＞0.由定理3可知 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459511&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459511&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459511&type=middle 从而对 j =1, 2, …, 有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459513&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459513&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459513&type=middle 另外，对 j ≤0同样也有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459520&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459520&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459520&type=middle 综上可知：如果 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459717&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459717&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459717&type=middle ，只要 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459719&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459719&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459719&type=middle 满足(15)式，那么对任意的 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459722&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459722&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459722&type=middle 有(16)式成立.因此如果 Φ ={ φ 1 , φ 2 , …, φ l }并且 Φ 满足(15)式，那么 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459524&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459524&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459524&type=middle 令 Φ = Ψ ，结合 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459726&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459726&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459726&type=middle 的定义便知 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459730&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459730&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459730&type=middle . 由定理4和定理5可得到以下结果: 定理6   令0＜ α ＜1， Ψ ={ ψ 1 , ψ 2 , …, ψ l }.设 X ( Ψ )是一个紧小波框架.如果 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459734&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459734&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459734&type=middle ， ψ ∈ Ψ ，那么 f ∈ Λ α 当且仅当 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459738&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459738&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459738&type=middle 且满足 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459530&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459530&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459530&type=middle 其中 C Ψ 为与 Ψ 有关的常数. Zygmund函数类 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459741&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459741&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459741&type=middle 的刻画 1 本节将给出Zygmund函数类 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459746&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459746&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459746&type=middle 的一个紧小波框架系数刻画. 文献[ 3 3 ]给出了Zygmund函数类 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459749&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459749&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459749&type=middle 的一个标准正交小波基系数刻画如下： 定理7   令 φ 为一个标准正交小波.如果 φ 属于正则类 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459752&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459752&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459752&type=middle ，那么 f ∈ Λ * 当且仅当 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459755&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459755&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459755&type=middle 且满足 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459533&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459533&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459533&type=middle 其中 C φ 为与 φ 有关的常数. 下面定理是本节的主要结果： 定理8   令 Ψ ={ ψ 1 , ψ 2 , …, ψ l }.设 X ( Ψ )是一个紧小波框架.如果 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459758&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459758&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459758&type=middle , ψ ∈ Ψ ，那么 f ∈ Λ * 当且仅当 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459760&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459760&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459760&type=middle 且满足 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459538&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459538&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459538&type=middle 其中 C ψ 为与 ψ 有关的常数. 证   只需证明(18)式和(19)式等价即可.在证明过程中设 X ( Ψ )是一个规范紧小波框架. 先证明“ (18)式推导(19)式”.设(18)式成立.则当 j ≤0时(18)式仍然成立.事实上， http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459542&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459542&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459542&type=middle 因此， http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459762&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459762&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459762&type=middle ，有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459546&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459546&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459546&type=middle 下证(19)式成立.因为 φ 为一个标准正交小波，所以 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459764&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459764&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459764&type=middle 和 j =1, 2, …, 有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459550&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459550&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459550&type=middle 因此由(20)式可得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459554&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459554&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459554&type=middle 估计|〈 ψ j , k , φ t , m 〉|的值.由定理2和定理3可知 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459558&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459558&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459558&type=middle 因此 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459561&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459561&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459561&type=middle 综上可知(19)式成立.如果设 X ( Ψ )是一个规范紧小波框架，那么(19)式推导(18)式的方法相同. 参考文献 SCHMIDT E. 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