微分算子与子波构造

将传统意义下的整数阶微分运算拓展到非整数阶微分情形,直接仿照整数阶微分在时域的极限定义形式是很困难的.本文从分析微分运算的频域形式着手,将微分算子分解成幅度算子和相位算子,并将其与子波变换特征进行比照研究,从而解决了非整数阶微分的拓展问题,同时也得到了微分运算与子波变换的内在联系,为非整数阶微分计算提供了一种逼近方式.文中提出了幅度算子、广义Hilbert变换等新概念并重点探讨了基于非整数阶微分运算的子波构造及其局域化特征等问题.

Differential Operator and the Construction of Wavelet

This paper investigates the topic of non integral order differentiation.Following directly the limit definition of the classical integral order differentiation in the time domain,it is difficult to expand the classical differential operation from the integral order case to the non integral order case.Based on the expression of differential operation in the frequency domain,and wedding it to the wavelet transform,the non integral order differential problem was solved in this paper.Meanwhile the inherent law between the non integral order differential operation and the wavelet transform was discovered,and an approximate approach for computing the non integral order differentiation was given.In this paper we introduce some new concepts,such as the magnitude operator and the generalized Hilbert transform etc,thereby study emphatically the construction and the localization characteristics of wavelet which is based on the non integral order differentiation of a low pass function.