Pairs of Dual Wavelet Frames fromAny Two Refinable Functions

Starting from any two compactly supported refinable functions in L₂(R)with dilation factor d,we show that it is always possible to construct 2d wavelet functionswith compact support such that they generate a pair of dual d-wavelet frames in L₂(R).Moreover, the number of vanishing moments of each of these wavelet frames is equalto the approximation order of the dual MRA; this is the highest possible. In particular,when we consider symmetric refinable functions, the constructed dual wavelets are alsosymmetric or antisymmetric. As a consequence, for any compactly supported refinablefunction in L₂(R), it is possible to construct, explicitly and easily, wavelets that arefinite linear combinations of translates (d · – k), and that generate a wavelet frame withan arbitrarily preassigned number of vanishing moments.We illustrate the general theoryby examples of such pairs of dual wavelet frames derived from B-spline functions.