Box Splines

My research began with this project which was initiated by paper [4].  I solved a conjecture related to the algebra of box spline space in [4] by introducing a concept of s-additivity and proving the equivalency between the conjecture and s-additivity. I learned how to conduct research in progress with this project by collaborating with Amos, Carl, Rong-Qing, and Sherm whom I regard as mentors of mine.  They developed my research capability and inspired me to become a mathematician.   I enjoyed my wonderful years as a Ph.D. student at the University of Alberta and a Research Associate at UW-Madison.

    1. Carl de Boor, Amos Ron, Zuowei Shen, On ascertaining inductively the dimension of the joint kernel of certain commuting linear operators II, Advances in Mathematics, 123 (1996), 223-242. PDF
    2. Carl de Boor, Amos Ron, Zuowei Shen, On ascertaining inductively the dimension of the joint kernel of certain commuting linear operators, Advances in Applied Mathematics, 17(1996), 209-250. PDF
    3. Rong Qing Jia, Sherman D. Riemenschneider, Zuowei Shen, Dimension of kernels of linear operators, American Journal of Mathematics 114(1992), 157-184. PDF
    4. Zuowei Shen, Dimension of certain kernel spaces of linear operators, Proceedings of the American Mathematical Society 112(1991), 381-390. PDF

Interpolaitions by Box Spliens

Hermite interpolation on the lattice was first introduced in [2]. Compactly supported fundamental solutions (Lagrange functions) for cardinal interpolations and Hermite interpolations were constructed in [1] using box splines. Approximation of fitting a curve, surface or function to scattered, possibly noisy, data can be found here.

    1. S. D. Riemenschneider, Zuowei Shen, Interpolation on the lattice hZs: Compactly supported fundamental solutions, Numerische Mathematik, 70(1995), 331-351. PDF
    2. K. Jetter, S.D. Riemenschneider, Zuowei Shen, Hermite interpolation on the lattice Zd, SIAM Journal on Mathematical Analysis 25(1994), 962-975. PDF