Short time Fourier transform, integral wavelet transform, and wavelet functions associated with splines (Q1270903)

The author considers a function of the form \(s_m=\sum_j c_j N_m(x-j)\), \(x\in{\mathbb R}\), \(j\in{\mathbb Z}\), where \((c_j)\in\ell_2\) and \(N_m(x)\) is the cardinal B-spline of order \(m\). First \(s_m\) is used as a kernel function in a short time Fourier transform. The time-frequency window can be modulated by the coefficients \((c_j)\). Next, conditions on the \((c_j)\) are derived to make \(s_m\) satisfy the admissibility condition so that it can be used as a wavelet function for an integral (i.e., continuous) wavelet transform. Using the two-scale relation for the \(N_m\), a (bi)orthogonal multiresolution analysis is then derived where the \(s_m\) act as (primal) scaling function. The corresponding wavelet functions are generalizations of B-wavelets. Finally, a recurrence relation is given for the computation of B-wavelets, and hence also for the wavelets considered in this paper.