A remark on vanishing moments

It is well known that with more vanishing moments on the scaling function and the wavelet function, one may get a better wavelet sampling approximation. In this paper, we present a quantitative treatment of the above statement. It is shown that equal vanishing moments distribution between the scaling function and the wavelet function is an ideal choice for the wavelet sampling approximation. Based on this observation, we introduce biorthogonal Coifman wavelet systems, a family of biorthogonal wavelet systems with the vanishing of moments equally distributed between scaling functions and wavelet functions. These wavelet systems provide a good wavelet sampling approximation with exponential decay, and the scaling vectors are all dyadic rational.