The Image Space of One Type of Continuous Wavelet Transform and Its Property

Abstract

In this paper, we show that the space of continuous wavelet transform is a reproducing kernel Hilbert space based on the fundamental theorem of linear transform. An admissible wavelet is got by convolution computation which is made into continuous wavelet transform. By the theory of reproducing kernel we can discuss correlative properties of image space of wavelet transform, which provide theoretic frame for us to study image space of the general wavelet transform.branch of mathematics. Its theory and methods also need to gradually be completed. Therefore, exploring new theories, new methods and new applications of wavelet analysis have become a hot issue of multi-disciplinary. 5 Reproducing kernel space is closed with the image space of the wavelet transform, because wavelet analysis is based on continuous wavelet transform, and continuous wavelet transform is based on reproducing kernel Hilbert space. The reproducing kernel Hilbert space plays an important role for the reconstruction of the continuous wavelet transform. 6,7 With the development of the theory of wavelet analysis, reproducing kernel theory aroused more attention of scholars, and its results is found endlessly. For example, there are some results by using the reproducing kernel theory to discuss the property of the image space of wavelet transform. Deng Caixia, Du Weiwei have described the image space of wavelet transform like Shannon, Marr, Littlewood-Paley and so on 8-10 ; Han Hong, Deng Caixia, Deng Zhongxing have described the image space of the improved Morlet wavelet transform 11 ; Deng Caixia, Qu Yuling, Hou Jie, Fu Zuoxian etc. made a more in-depth study of the image space of Gauss and Gabor etc. wavelet transforms. 12-16 It can be seen that the above results are taken into account the property of the image space of wavelet transform which are commonly used to the typical wavelet functions. In this paper, the admissible wavelet which is obtained by convolution is more flexibility, and the research of the property of image space of the wavelet transform also has more general theoretical and practical application value, and the results of Ref. 13 have become a special case of this paper. The Definition and Properties of Reproducing KernelLet F (E) be a linear space comprising all complex-valued functions on an abstract set E. Let H be a Hilbert space equipped with inner product (·, ·) H .