一个二进小波变换像空间的再生核函数

1 引言小波分析是结合泛函分析、应用数学、逼近论、调和分析、广义函数论等数学知识的结晶,具有深刻的理论意义和广泛的应用范围,被称为"数学显微镜".基于其多分辨分析的特点以及在时、频两域都具有表征信号局部特征的功能,应用它可以解决许多Fourier变换不能解决的难题,为工程应用提供了一种新的、更有效的分析工具1].

THE REPRODUCING KERNEL FUNCTION OF IMAGE SPACE OF A DYADIC WAVELET TRANSFORM

Firstly,this paper constructs a dyadic wavelet with non-orthogonality,symmetry,limited spectrum and full smooth almost everywhere,discusses the dual wavelet and its properties,and gets the isometric and the estimation of the dyadic wavelet transform.Secondly,the analytic expression of the re-producing kernel function of image space of the dyadic wavelet transform is obtained by using the reproducing kernel Hilbert space theory.When the scale factor is fixed,the reproducing kernel function of image space of the dyadic wavelet transform is further characterized.Reproducing kernel illustrates the redundancy of wavelet transform so that the dyadic wavelet transform can be represented by reproducing kernel.Finally different forms of expansion of dyadic wavelet transform are given,and the inverse formula of dyadic wavelet transform are obtained by using dyadic dual wavelet.The proposed theorems not only provide a reference for the study of image space of general wavelet transform,but also provide a theoretical basis for the numerical calculation of wavelet in engineering applications.