Starting from any two compactly supported refinable functions in L 2 (R) with dilation factor d, we show that it is always possible to construct 2d wavelet functions with compact support such that they generate a pair of dual d-wavelet frames in L 2 (R). Moreover, the number of vanishing moments of each of these wavelet frames is equal to 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 also symmetric or antisymmetric. As a consequence, for any compactly supported refinable function in L 2 (R), it is possible to construct explicitly and easily wavelets that are finite linear combinations of translates (d k), and that generate a wavelet frame with arbitrarily preassigned number of vanishing moments. We illustrate the general theory by examples of such pairs of dual wavelet frames derived from B-spline functions.

Abstract. Let φ be a compactly supported symmetric real-valued refinable function in L2(R) with a finitely supported symmetric real-valued mask on Z. Under the assumption that the shifts of φ are stable, in this paper we prove that one can always construct three wavelet functions ψ1, ψ2 and ψ3 such that (i) All the wavelet functions ψ1, ψ2 and ψ3 are compactly supported, real-valued and finite linear combinations of the functions φ(2 · −k), k ∈ Z; (ii) Each of the wavelet functions ψ1, ψ2 and ψ3 is either symmetric or antisymmetric; (iii) {ψ1, ψ2, ψ3} generates a tight wavelet frame in L2(R), that is,

Abstract. An interesting method called Oblique Extension Principle (OEP) has been proposed in the literature for constructing compactly supported MRA tight and dual wavelet frames with high vanishing moments and high frame approximation orders. Many compactly supported MRA wavelet frames have been recently constructed from scalar refinable functions via OEP. Despite the great flexibility and popularity of OEP for constructing compactly supported MRA wavelet frames in the literature, however, the associated fast frame transform is generally not compact and a deconvolution appears in the frame transform. Here we say that a frame transform is compact if it can be implemented by convolutions, coupled with upsampling and downsampling, using only finite-impulse-response (FIR) filters. In this paper we shall address several fundamental issues on MRA dual wavelet frames and fast frame transforms. Basically, we present two complementary results on dual wavelet frames which are obtained via OEP from scalar refinable functions (=refinable function vectors with multiplicity one) and from truly refinable function vectors with multiplicity greater than one. On the one hand, by a nontrivial argument, we show that from any pair of compactly supported refinable spline functions φ and ˜ φ (not necessarily having stable integer shifts) with finitely