Cubic spline wavelet bases of Sobolev spaces and multilevel interpolation

In this paper, a semi-orthogonal cubic spline wavelet basis of homogeneous Sobolev space H-0(2)(I) is constructed, which turns out to be a basis of the continuous space C-0(I). At the same time, the orthogonal projections on the wavelet subspaces in H-0(2)(I) are extended to the interpolating operators on the corresponding wavelet subspaces in C-0(I). A fast discrete wavelet transform (FWT) for functions in C-0(I) is also given, which is different from the pyramid algorithm and easy to perform using a parallel algorithm. Finally, it is shown that the singularities of a function can be traced from its wavelet coefficients, which provide an adaptive approximation scheme allowing us to reduce the operation time in computation. (C) 1996 Academic Press, Inc.