Multiresolution Analysis On Two-dimensional Cantor Sets And Research On Wavelet-related Problems Based On Vilenkin Group

As early as 1807,Fourier that was a famous French mathematician and physicist proposed a widely theory applied to every field in the future.It is Fourier transform.It took more than a century as much as before gradually mature from 1807 to 1966.It studies complex functions by analyzing functions.It not only plays an important role in physics,but also has an immeasurable role in every branch of Mathematics.But the Fourier transform also has disadvantages,such as the result of Fourier transform processing the non-stationary signal is not consistent with the actual result in real life.Gabor discovered the disadvantages of Fourier transform and proposed a windowed Fourier transform.Although the windowed Fourier transform can solve some problems of the Fourier transform,it can not deal with the singular signals.Although the windowed Fourier transform is better than the Fourier transform,but its shortcomings still need to make improvements.Wavelet transform emerged at a historic moment.The wavelet transform is developed on the basis of Fourier transform and windowed Fourier transform,but it can overcome the shortcomings of Fourier transform and windowed Fourier transform.The wavelet basis function ?a,b b(t)is equivalent to the windowed function and is variable.As long as the function satisfies certain conditions,it can be used as a wavelet basis function.So how to find a satisfactory wavelet basis function is a very important problem in wavelet transform.In 1988,Mallat and Meyer established a multiresolution analysis which provided a method for constructing wavelet basis functions.It is gradually developed and tends to improve under the joint efforts of scholars with different backgrounds in the process of the development of wavelet analysis.It promoted the development and progress in many fields of the world in the meantime.The discovery of wavelet analysis theory fully illustrates varied disciplines mutual integration and mutual infiltration in the era of big science.The main contents of this paper are as follows:Chapter one:Preliminary knowledge.The relevant notations and the related know-ledge of Fourier transform and wavelet analysis are introduced.Chapter two:The MRA of two-dimensional Cantor set.According to the concept of multiresolution analysis,the multiresolution analysis on the two-dimensional Cantor set is given,which can make the readers better understand the multi-resolution analysis.Chapter three:The Algorithm for Wavelets on the p-ddic Vilenkin group.The length-4 scaling filters and the Mallat tree-type algorithm on the p-adic Vilenkin group-Chapter four:Wavelet methods for differential equations on the p-adic Vilenkin gr-oup.The Fourier-Walsh series on the p-adic Vilenkin group is introduced.The existence and uniqueness of the solution of the linear differential equation of order n and the one-dimensional homogeneous wave equation are given.