Dual Wavelet Frames and Riesz Bases in Sobolev Spaces

This paper generalizes the mixed extension principle in L-2(R-d) of (Ron and Shen in J. Fourier Anal. Appl. 3: 617-637, 1997) to a pair of dual Sobolev spaces H-s ( Rd) and H-s (R-d). In terms of masks for phi, psi(1),...,psi(L) is an element of H-s (R-d) and (phi) over tilde, (psi) over tilde (1),..., (psi) over tilde (L) is an element of H-s (R-d), simple sufficient conditions are given to ensure that (X-s(phi; psi(1),...,psi(L)), X-s ((phi) over tilde; (psi) over tilde (1),..., (psi) over tilde (L))) forms a pair of dual wavelet frames in (H-s(R-d), H-s (R-d)), where X-s (phi; psi(1),...,psi(L)) := {phi (.-k) : k is an element of Z(d)} boolean OR{2(j(d/2-s))psi(l)(2(j).-k) : j is an element of N-0, k is an element of Z(d), l = 1, ... , L}. For s > 0, the key of this general mixed extension principle is the regularity of phi, psi(1),..., psi(L), and the vanishing moments of (psi) over tilde (1),..., (psi) over tilde (L), while allowing (phi) over tilde, (psi) over tilde (1),..., (psi) over tilde (L) to be tempered distributions not in L-2(R-d) and psi(1),..., psi(L) to have no vanishing moments. So, the systems X-s(phi; psi(1),..., psi(L)) and X-s ((phi) over tilde; (psi) over tilde (1),..., (psi) over tilde (L)) may not be able to be normalized into a frame of L-2(R-d). As an example, we show that {2(j(1/2-s)) B-m(2(j).-k) : j is an element of N-0, k is an element of Z} is a wavelet frame in H-s (R) for any 0 < s < m-1/2, where B-m is the B-spline of order m. This simple construction is also applied to multivariate box splines to obtain wavelet frames with short supports, noting that it is hard to construct nonseparable multivariate wavelet frames with small supports. Applying this general mixed extension principle, we obtain and characterize dual Riesz bases (X-s(phi; psi(1),..., psi(L)), X-s ((phi) over tilde; (psi) over tilde (1),..., (psi) over tilde (L))) in Sobolev spaces (H-s(R-d), H-s (R-d)). For example, all interpolatory wavelet systems in (Donoho, Interpolating wavelet transform. Preprint, 1997) generated by an interpolatory refinable function phi is an element of H-s (R) with s > 1/2 are Riesz bases of the Sobolev space H-s (R). This general mixed extension principle also naturally leads to a characterization of the Sobolev norm of a function in terms of weighted norm of its wavelet coefficient sequence (decomposition sequence) without requiring that dual wavelet frames should be in L-2( R-d), which is quite different from other approaches in the literature.