ON THE ASYMPTOTIC CONVERGENCE OF B-SPLINE WAVELETS TO GABOR FUNCTIONS

A family of nonorthogonal polynomial spline wavelet transforms is considered. These transforms are fully reversible and can be implemented efficiently. The corresponding wavelet functions have a compact support. It is proven that these B-spline wavelets converge to Gabor functions (modulated Gaussian) pointwise and in all L(p)-norms with 1 less-than-or-equal-to p < + infinity as the order of the spline (n) tends to infinity. In fact, the approximation error for the cubic B-spline wavelet (n = 3) is already less than 3%; this function is also near optimal in terms of its time/frequency localization in the sense that its variance product is within 2% of the limit specified by the uncertainty principle.