Dual Wavelet Frames and Riesz Bases in Sobolev Spaces

Abstract

Abstract. This paper generalizes the mixed extension principle in L 2 (R For s > 0, the key of this general mixed extension principle is the regularity of , k ∈ 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 (. For example, all interpolatory wavelet systems in [25] generated by an interpolatory refinable function φ ∈ 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 to other approaches in the literature.

Linearized Bregman Iterations for Frame-Based Image Deblurring

“…The second term is to penalize the distance of u to the range of F , so it makes u close to its canonical tight frame coefficient. By the theory in [3,34], the weighted 1 norm of the canonical coefficient is related to the Besov norm of the underlying solution when F is a tight wavelet frame (or so-called framelet) system. Therefore, these two terms together balance the sparsity of the tight frame coefficient and the regularity (the smoothness) of the underlying solution that has the same flavor of the algorithm given in [7,8,12].…”

Tight Frame Based Method for High-Resolution Image Reconstruction

“…The second term measures the distance from α to the range of A. By the framelet theory in [2,24], if the coefficient α is in the range of A, then the (weighted) 1 norm α is equivalent to some Besov norm of the image A T α. Therefore, the second term links the (weighted) 1 norm to the regularity of the restored image in some sense.…”

Characterization of Sobolev spaces of arbitrary smoothness using nonstationary tight wavelet frames

“…In our recent work [16], we have systematically studied the construction of a pair of dual wavelet frames in a pair of dual Sobolev spaces, which is not necessarily a pair of dual frames in L 2 (R). We also used them to characterize the Sobolev norm of functions in terms of the weighted 2 -norm of the analysis wavelet coefficient sequences in [16].…”

“…In our recent work [16], we have systematically studied the construction of a pair of dual wavelet frames in a pair of dual Sobolev spaces, which is not necessarily a pair of dual frames in L 2 (R). We also used them to characterize the Sobolev norm of functions in terms of the weighted 2 -norm of the analysis wavelet coefficient sequences in [16]. However, since there is no stationary compactly supported wavelet frame with the infinite smoothness order, it is impossible to characterize all Sobolev spaces with one fixed stationary compactly supported wavelet frame.…”