Computation of Battle-Lemarie wavelets using an FFT-based algorithm

Battle and Lemarie [1,2] derived independently wavelets by orthonormalizing B-splines. The scaling function phi(m)(t) corresponding to Battle-Lemarie's wavelet psi(m)(t) is given by phi(m)(t) = Sigma(k is an element of Z)alpha(m,k)B(m)(t-k), where B-m(t) is the mth-order central B-spline. In this paper, we propose an FFT-based algorithm for computing the expansion coefficients alpha(m,k) and the two-scale relations of the scaling functions and wavelets. The algorithm is very simple and it can be easily implemented. Moreover, the expansion coefficients can be efficiently and accurately obtained via multiple sets of FFT computations.