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Research interests My research interests are centered
around the approximation theory, computational harmonic analysis, numerical
analysis and discrete mathematics. Topics that I am working on include
Box splines
and algebra
Box splines are tools in multivariate approximation theory. It is also related with combinatorics and algebra. In particular, it plays a key role in investigating the number of integer solutions of linear Diophantine equations,the number of integer points in polytopes and the volume of polytopes. Moreover, zonotopal space also arises from box splines. Selected
papers: l
Hierarchical zonotopal spaces, with
O. Holtz and A. Ron, Trans. Amer. Math. Soc., Volume 364, Number 2, 2012. l Multivariate splines and polytopes, Journal of Approximation Theory, Vol. 163, Issue
3, March 2011. l Sagbi bases of Cox-Nagata
rings, with Bernd Sturmfels, Journal of the European Mathematical
Society, Volume 12, Issue 2, 2010. l
Eulerian
numbers: a spline interpretation ,With
R. Wang, Y. Xu, J. Mathematical Analysis and Applications, 370 (2010) 486–490. l
Multivariate
F-splines and fractional box splines,
Journal of Fourier Analysis and Applications,
15:723-738, 2009. l
Multi-dimensional versions of
a formula of Popoviciu, Science in China Series A. , 2006. l
Discrete Truncated
Power And Lattice Points In Rational Polytope , with
Ren-Hong Wang, Journal of Computational and Applied Mathematics 159 (2003)
149-159. l
Refinement equations and spline functions, with
A. Dubickas, Adv. Comp. Math. 32: 1-23, 2010. l Marginal
Likelihood Integrals for Mixtures of Independence Models, with
Shaowei Lin and Bernd Sturmfels,
Journal of Machine Learning Research, 10(Jul):1611--1631, 2009. l The regularity of refinable functions, with Yang Wang,
Applied Computational Harmonic Analysis，Volume
34, Issue 1, Pages 1-162, January
2013. Compressed
sensing and related problems
One can recover the sparse signal from a few measurement Samples. To do
that, we need construct encoding matrix and decoding algorithm carefully.
Random matrixes play a key role in constructing encoding matrix in compressed
sensing. I am interested in
constructing the deterministic encoding matrix. I also like some conjectures
related with random matrix. Selected
papers: l
One-Bit Compressed Sensing by Greedy
Algorithms, with Wenhui Liu, D. Gong l
Phase Retrieval for Sparse Signals， with Yang Wang. l A
strong restricted isometry property, with an application to phaseless
compressed sensing, with V. Voroninski, submitted for publication. l Robustness
Properties of Dimensionality Reduction with Gaussian Random Matrices,
with Bin Han. l
On sparse interpolation and the
design of deterministic interpolation points, with T. Zhou. l
On the $\ell_1$-Norm Invariant Convex
k-Sparse Decomposition of Signals， with G. Xu l
Compressed Sensing Matrices from
Fourier Matrices, with G. Xu,. l The performance of orthogonal multi-matching pursuit under RIP , l
Compressed
sensing, A
survey in Chinese,
Sci Sin Math, 2012, 42(9). l
Deterministic
Sampling of Sparse Trigonometric Polynomials, Journal of
Complexity, Volume 27, Issue
2, April 2011, Pages 133-140. l A remark about orthogonal matching
pursuit algorithm, Advances Adaptive Data Analysis, 2013. Frame theory and quantization
Frames become more and more popular in applied mathematics, in particular in image process, information theory etc. I am interested in B-spline wavelet frames and construct maximally equiangular frames. And also apply frames to quantization. Selected papers:
l The lower bound of the PCM quantization error in high dimension, with H. Zhou, Applied Computational Harmonic Analysis, 2015 l On B-spline framelets derived from the unitary extension principle, with Zuowei Shen, SIAM Journal on Mathematical Analysis, 45(11) 2013, 127-151. l The Performance of PCM Quantization Under Tight Frame Representations, with Yang Wang, SIAM J. MATH. ANAL, Vol. 44, No. 4, pp. 2802–2823, 2012. l
Adaptive
non-uniform B-spline dictionaries on a compact interval, With Laura
Rebollo-Neira, Signal Processing,
Spectral
set conjecture
Recently, I am interested in spectral set conjecture, which was from Fuglede (1974). The conjecture is related with wavelet and sampling theory. It was disproved in dimensions 3 and higher by Tao, Kolountzakis, Matolcsi, Farkas and Mora. But it remains open in dimensions 1 and 2. Multivariate
splines and computer aided geometry design
I
worked on this topics before 2008. Selected
papers: l Discrete Schemes for Gaussian Curvature
and Their Convergence, with Guoliang Xu, Computers and Mathematics with
Applications, 2009. l Convergence analysis of discrete differential geometry operators
over surfaces, Lecture Notes in Computer Science, Mathematics of
Surfaces XI, 2005. l Analytic and algebraic properties of canal surfaces,
Journal of Computational and Applied Mathematics, 195(2006)(With Jiaguang
Sun, etc). l A robust algorithm for finding the real intersections of
three quadric surfaces, Computer Aided Geometric Design, Vol 22, Issue 6,
1 (2005), 515-530. l The Structural Characterization and Locally Supported Bases
for Bivariate Super Splines, with Ren-hong Wang, Journal of
Computational Mathematics.6(2004). l The
Estimation of the Bezout Number of Piecewise Algebraic Curve, with
Ren-hong Wang, Science in China Series A, 2003 Vol.46 No.5, 710-717. l The instability degree in the dimension of spaces of bivariate spline, with Ren-hong Wang, Approx. Theory & its Appl., 18:1,2002,68-80. |