Functions On The Vilenkin Group And Their Representations

Wavelet analysis is always called mathematical microscope.When applying wavelets to analysis signal,it can make the signal have the local characters on both time and frequency,and carries on the multi-resolution analysis.Processing the dis-crete data problems becomes relatively easy because of this property.Then,wavelet analysis has been widely applied in digital signal analysis and image processing,such as:filtering,classification,identification,transmission,denoising,compression,diagnosis and so on.In the nature,there are many irregular things whose shape has the self-similarity between its entirety and locality,thus,applying the fractal theory to simulate their complexity is very suitable.So combing fractal and wavelet theory plays an important role in solving problems in real life.This article mainly studied the algorithm of signal decomposition and reconstruction on fractal set,to lay a solid theoretical foundation for its application in the future.At present,advanced scholars have explored orthogonal wavelets,wavelet frames and the algorithm of their construction,and enhanced the theory of the multi-resolution analysis on p-adic Vilenkin groups.Based on the summarization of the existing research results of wavelet functions,this paper includes the Fourier-Walsh transformation on the p-adic Vilenken groups,and chooses Haar wavelet,which has the positive quality on symmetry,compactly supported,othogonality and so on,to detail the theory of the multi-resolution analysis on it.In the aspect of processing signal,we studied the algorithm of signal decomposition and reconstruction,in order to understand the local features of signal better and extract some useful informa-tion more quickly.We followed on to generalize it to the two-dimensional space,discussing the theory of two-dimensional multi-resolution analysis and applying the two-dimensional wavelet transform to decompose signal,the two-dimensional inverse wavelet transform to reconstruct signal.Only by understanding the the unique nature of the signal can you deal with the signal better and solve the practical problems in life easily.In this paper,we shot a light on the based on 1-periodic locally constant functions,step functions,local-ly constant compactly supported functions respectively,and introduced its nature.Then,we studied 1-periodic distributions and its walsh representation,step distri-butions and its step representation,and tempered distributions and its quasi-Haar representation in depth.