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

Battle and Lemarie (1987, 1988) derived independently wavelets by orthonormalizing B-splines. The scaling function φ_m(t) corresponding to Battle-Lemarie´s wavelet ψ_m(t) is given by φ_m(t)=Σ_k∈Zα_m,kB _m(t-k), where B_m(t) is the mth-order central B-spline. We propose an FFT-based algorithm for computing the expansion coefficients α_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