Notes on wavelet-like basis matrices

In this paper, we improve the algorithms for the construction of the wavelet-like basis matrix introduced by Alpert et al. [1]. It has been shown in [1] that the n × n wavelet-like basis matrix is of the form U = UlUl−1…U1, where n = k2l is the number of quadrature points and Uj, J = 1…,l are sparse orthogonal matrices. In this paper, we prove that each Uj (1 ≤ j ≤ l) can be represented by a 2k × 2k matrix. It follows that the storage requirement for all matrices Uj is 4lk2. We also show that the cost of the construction of all matrices Uj can be reduced to O(lk3) = O(logn • k3). We recall that in [1], the storage requirement and the construction cost of the matrix U are 4nk and O(nk2), respectively.