Constructions using Box Splines

We started the topic of constructions of multivariate wavelets using box splines in [10] by constructing orthogonal wavelets with exponential decay. After this, we designed compactly supported pre-wavelet in [9], wavelets from the loop scheme in [3], and wavelets of small support in [1] using box splines that have dual systems with infinite supports. Such wavelets are used in applications as computer graphics where only one of the analysis\synthesis operators is needed. When both analysis and synthesis operators are used, we need compactly supported wavelet systems with a compactly supported dual system. This led to the compactly supported multivariate biorthogonal wavelets in [4, 5]. The key step is to construct interpolatory subdivision schemes. To analyze the interpolatory subdivision scheme of [6], we derived the characterizations properties of refinable functions in terms of refinement masks,

The wavelet systems discussed here are non-redundant, for example, orthonormal or Riesz wavelet systems., whose constructions are not easy,. However, constructions of redundant systems, such as tight frames and bi-frames, are much simpler thanks to the extension principles.

      1. Bin Han, Qun Mo, Zuowei Shen, Small support spline Riesz wavelets in low dimensions, Journal of Fourier Analysis and Applications, 17(4), (2011), 535-566. PDF
      2. Bin Han, Zuowei Shen, Wavelets with short support, SIAM Journal on Mathematical Analysis, 38(2) (2006), 530-556. s_wavelet.pdf
      3. Bin Han, Zuowei Shen, Wavelets from the Loop scheme, Journal of Fourier Analysis and Applications, 11(6) (2005), 615-637. loop.pdf
      4. Hui Ji, S. D. Riemenschneider, Zuowei Shen, Multivariate compactly supported fundamental refinable functions, duals, and biorthogonal wavelets, Studies in Applied Mathematics,102 (1999), 173-204. PDF
      5. S.D. Riemenschneider, Zuowei Shen, Construction of compactly supported biorthogonal wavelets in $L_2(R^d)$ II, Wavelet applications signal and Image Processing VII} Proceedings of SPIE Volume 3813, (1999), Michael A. Unser, Akram Aldroubi, and Andrew F. Lain eds, 264-272. PDF
      6. S. D. Riemenschneider and Zuowei Shen, Multidimensional interpolatory subdivision schemes, SIAM Journal on Numerical Analysis, 34 (1997), 2357-2381. PDF
      7. Zuowei Shen, Non-tensor product wavelet packets in L2( Rs), SIAM Journal on Mathematical Analysis, 26(1995), 1061-1074.
      8. Rong Qing Jia, Zuowei Shen, Multiresolution and wavelets, Proceedings of the Edinburgh Mathematical Society 37(1994), 271-300. PDF
      9. S. D. Riemenschneider, Zuowei Shen, Wavelets and pre-wavelets in low dimensions, Journal of Approximation Theory 71(1992), 18-38. PDF
      10. S. D. Riemenschneider, Zuowei Shen, Box splines, cardinal series, and wavelets, in Approximation Theory and Functional Analysis, C.K. Chui eds., Academic Press, New York, (1991), 133-149.