Wavelet Decomposition Associated With Grushin Operator

In this paper,we construct orthonormal wavelet bases on two-dimensional Euclidean space,and extend the results to n-dimensional Euclidean space.At the same time,we establish orthonormal wavelet bases on two-dimensional Euclidean space which is related to grushin operator,and give the decomposition of functions in wavelet bases.The main contents are:Chapter 1 briefly introduces the wavelet,current situation of the wavelet.Chapter 2 gives the orthonormal wavelet bases on Euclidean space R2 where the wavelets supported on binary squares which have constant translation in the x and y directions and are of equal length.The classical wavelet decomposition of L2(R2)function is given.Chapter 3 gives the orthonormal wavelet bases on Euclidean space R2 and these wavelets are supported on binary cuboids which have unchanged translation but unequal length at any direction,then introduces the anisotropic function spaces Bpqs,?(Rn)and Fpqs,?(Rn)on Rn constructed by Triebel and the relationship with the wavelet bases mentioned above.Chapter 4 establishes the orthonormal wavelet bases on homogeneous space(R2,?,(?))associated with Grushin operator on Euclidean space R2.These wavelets are supported on binary rectangles,which have constant translation in the x direction and do not have translation-invariant properties in the y direction;at the same scale,the side length of the rectangle changes with the y coordinate of the center of the rectangle.Then we give the decomposition of L2(R2)function in these wavelet bases.