Constructions using Box Splines

The multivariate wavelets started from [10] where we constructed orthonormal wavelets using box splines. Then, we designed compactly supported pre-wavelet in [9], wavelets from the loop scheme in [3], and wavelets with small support in [1] using box splines. Such wavelets were used in applications such as computer graphics where only one of the analysis\synthese operators is needed. When both analysis and synthesis operators are used, we need a compactly supported wavelet system with a compactly supported dual system. This motivated our work on compactly supported multivariate biorthogonal wavelets in [4, 5].  One of the key steps in constructing biorthogonal wavelets is to construct interpolatory subdivision schemes. The analysis of the interpolatory subdivision scheme constructed in [6] led to our work on the characterizations of properties of refinable functions in terms of refinement masks,

The wavelet systems discussed here are non-redundant multiresolution analysis based systems, i.e. orthonormal wavelet systems or  Riesz wavelet systems with dual wavelet systems. The constructions are not so simple, while the constructions of redundant multiresolution analysis based 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.