Cubic Spline Wavelet Bases of Sobolev Spaces and Multilevel Intorpolation

In this paper, it is constructed that a semi-orthogonal cubic spline wavelet basis of homogeneous Sobolev space H 2 0 (I), which turns out to be a basis of continuous space C 0 (I). Meanwhile, the orthogonal projections on the wavelet subspaces in H 2 0 (I) are extended to the interpolating operators on the corresponding wavelet subspaces in C 0 (I). It is also given that a fast discrete wavelet transform (FWT) for functions in C 0 (I) which is diierent from the pyramid algorithm and easy to perform by using parallel algorithm. Finally, it is shown that the singularities of a function can be traced from its wavelet coeecients, which provides an adaptive approximation scheme allowing us to reduce the operation time in computation.