Computation of battle-lemarie wavelets using an FFT-based algorithm

Battle and Lemarie derived independently wavelets by orthonormalizing B-splines. The scaling function Φm(t) corresponding to Battle-Lemarie's wavelet ψm(t) is given by Φm(t) = ∑kεZ αm,k Bm(t-k), where B m(t) is the mth-order central B-spline and the coefficients αm,k satisfy ∑kεZ αm,k e-fkw = 1/√∑kεZB2m(k)e -fkw. In this paper, we propose an FFT-based algorithm for computing the expansion coefficients αm,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. The computational approach presented in this paper here is noniterative and is more efficient than the matrix approach recently proposed in the literature.