L² Convergence of Biorthogonal Coifman Wavelet Systems

Biorthogonal Coifman wavelet (BCW) systems are biorthogonal wavelet systems with the vanishing of moments equally distributed between the scaling functions and wavelet functions. It has been shown that these wavelet systems provide an optimal wavelet sampling approximation with an exponential convergence rate, where the optimality is measured over all possible vanishing moments distributions. The scaling filters and wavelet filters are all dyadic rational, which means we can implement a very fast multiplication-free discrete wavelet transform. Recent work showed that the scaling filters of BCW systems converge to the sinc scaling filter in l²(Z), while the L²(R) convergence of the scaling functions remained open. In this work we will prove the scaling functions of BCW systems converges to the sinc function in L²(R). Following the pioneering work of Chui and Wang, we establish a quantitative relationship between the orthonormal Daubechies wavelet systems and the BCW sys..