Bibliography

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[109] M. J. Lai and J. Wang, An unconstrained lq minimization for sparse solution of under determined linear systems , submitted, 2009.

[108] M. J. Lai and J. Wang, Convergence of a Central Dierence Discretization of ROF Model for Image Denoising, submitted, 2009.

[107] S. Foucart and M. J. Lai, Sparse Recovery with Pre-Gaussian Random Matrices, submitted, 2009.

[106] S. Kersey and M. J. Lai, Nonuniform Variational Interpolatary Refinement , submitted, 2009.

[105] Guillas and M. J. Lai, Bivariate Splines for Spatial Functional Regression Models, to appear in Journal of Nonparametric Statistics, 2009.

[104] M. J. Lai and A. Petukhov, Method of Virtual Components in the Multivariate Setting, accepted for publication in J. of Fourier Analysis and Its Applications, 2009.

[103] M. J. Lai, On sparse solution of underdetermined linear systems, accepted for publication in Journal of Concrete and Applicable Mathematics, 2009,

[101] M. J. Lai and L. L. Schumaker, Domain decomposition method for scattered data fitting , SIAM J. Numerical Analysis, 47(2009), 911-928.

[100] S. Foucart and M. J. Lai, Sparsest Solutions of Underdetermined Linear Systems via \ell_q-minimization for 0\le q \le 1, , Applied and Computational Harmonic Analysis, 26(2009) 395--407.

[99] M. J. Lai, C. K. Shum, V. Baramidze, and P. Wenston, Triangulated spherical splines for geopotential reconstruction, J. of Geodesy, 83 (2009) 695--708.

[98] M. J. Lai and K. Nam, On the Number of Tight Wavelet Frame Generators associated with Multivariate Box Splines, Accepted for publication by Journal of Approximation Theory and Its Applications, 2008.

[97] M. J. Lai, Multivariate Splines and Their Applications, to appear in Encyclopedia of Complexity and System Science, edited by Meyers, Springer Verlag, 2008.

[96] M. J. Lai, Popular Wavelet Families and Filters and Their Use, to appear in Encyclopedia of Complexity and System Science, edited by Meyers, Springer Verlag, 2008.

[95] Scott Kersey and Ming-Jun Lai, Convergence of local variational spline interpolation , J. Math. Anal. Appl. 341 (2008) 398--415.

[94] Ming-Jun Lai, Multivariate splines for data fitting and approximation, Approximation Theory XII, San Antonio, 2007, edited by M. Neamtu and L. L. Schumaker, Nashboro Press, 2008, Brentwood, TN, pp. 210--228.

[93] Tian-he Zhou, Dan-Fu Han, Ming-Jun Lai, Energy Minimization Method for Scattered Data Hermite Interpolation, Applied Numerical Mathematics (2008) 58, 646--659.

[92] Xian-Liang Hu, Dan-Fu Han, Ming-Jun Lai, Bivariate Splines of Various Degrees for Numerical Solution of Partial Differential Equations, SIAM Journal of Scientific Computing, 29(2007) 1338--1354.

[91] M. J. Lai and L. L. Schumaker, Trivariate Cr polynomial macro-elements, Constructive Approxi. 26(2007), 11--28.

[90] M, J. Lai, Convergence of three $L_1$ spline methods for scattered data interpolation and fitting, , Journal of Approximation Theory 145(2007), 196--211.

[89] M. J. Lai and A. Petukhov, Method of virtual components for constructing redundant filter banks and wavelet frames, Applied and Computational Harmonic Analysis, 22(2007) 304--318.

[88] M. J. Lai and L. L. Schumaker, Spline Functions over Triangulations, Cambridge University Press, April 30, 2007.

[87] G. Chen and M. J. Lai, Wavelets and Splines: Athens, 2005, Nashboro Press, Brentwood, 2006.

[86] M. J. Lai, A. Le Mehaute, and T. Sorokina, An octohdral $C^2$ macro-element, Comp. Aided Geom. Design, 23(2006), 640--654.

[85] M. J. Lai, J. A. Lian, and P. Cassidy, Removal of Gaps among Compound C^1 Bi-Cubic Parametric B-spline Surfaces, in Wavelets and Splines: Athens 2005, edited by G. Chen and M. J. Lai, Nashboro Press, 2006, 287--314.

[84] G. Awanou, M. J. Lai, and P. Wenston, The multivariate spline method for numerical solution of partial differential equations and scattered data interpolation , in Wavelets and Splines: Athens 2005, edited by G. Chen and M. J. Lai, Nashboro Press, 2006, 24--74.

[83] M. J. Lai and K. Nam, Tight Wavelet Frames over Bounded Domains , in Wavelets and Splines: Athens, 2005, edited by G. Chen and M. J. Lai, Nashboro Press, 2006, pp. 313--326.

[82] M. J. Lai, Construction of multivariate compactly supported prewavelets in L_2 spaces and pre-Riesz basis in Sobolev spaces, Journal of Approximation Theory 142(2006), 83--115.

[81] O. Cho and M. J. Lai, A class of compactly supported orthonormal B-Spline wavelets, in Wavelets and Splines, edited by G. Chen and M. J. Lai, Nashboro Press, 2006, pp. 123--151.

[80] M. J. Lai and J. Stoeckler, Construction of multivariate compactly supported tight wavelet frames, Applied and Comput. Harmonic Analysis 21(2006), 324--348.

[79] V. Baramidze and M. J. Lai, Spherical Spline Solution to a PDE on the Sphere, in Wavelets and Splines, edited by G. Chen and M. J. Lai, Nashboro Press, 2006, 75--92.

[78] J. Geronimo and M. J. Lai, Factorization of Multivariate Positive Laurent Polynomials , Journal of Approximation Theory, 139(2006), 327--345.

[77] V. Baramidze, M. J. Lai, and C. K. Shum, Spherical Splines for Data Interpolation and Fitting, SIAM J. Scientific Computing, 28(2006), 241--259.

[76] M. J. Lai, Construction of multivariate compactly supported orthonormal wavelets , Advances in Computational Mathematics 25(2006), 41--56.

[75] X. Wang, H. Wang, and M. J. Lai, Some Results on Numerical Divided Difference Formulas, 2005. (See here for English version.) Scientia Sinica, Ser. A., 35(2005), pp. 712--720.

[74] V. Baramidze and M. J. Lai, Error Bounds for Minimal Energy Interpolatory Spherical Splines, Approximation Theory XI, edited by C. K. Chui, M. Neamtu, and L. L. Schumaker, Nashboro Press, Brentwood, 2005, pp. 25--50.

[73] G. Awanou and M. J. Lai, On the convergence rate of the augumented Lagrangian algorithm for the nonsymmetric saddle point problem, J. Applied Numerical Mathematics, (54) 2005, 122--134.

[72] G. Awanou and M. J. Lai, Trivariate Spline Approximations of 3D Navier-Stokes Equations , Mathematics of Computation, 74(2005), 585--601.

[71] V. Baramidze and M. J. Lai, Volume data interpolation by tensor products of spherical and radial splines , in Advances in Construtive Approximations, edited by M. Neamtu and E. Saff, Nashboro Press, 2004, pp. 75--88.

[70] X. H. Wang, M. J. Lai, and S. Yang, On Divided Differences of the Remainder of Hermite Interpolation Polynomial , Journal of Approximation Theory, (127) 2004, pp. 193--197.

[69] M. J. Lai, Chun Liu, and Paul Wenston, On two nonlinear biharmonic evolution equations, Applicable Analysis 83(2004), 541--562.

[68] M. J. Lai, Chun Liu, and Paul Wenston, Numerical simulation of two nonlinear biharmonic equations, Applicable Analysis 83(2004), 562--577.

[67] M. J. Lai and A. LeMehaute, A new kind of trivariate $C^1$ finite element , Advances in Computational Math. 21(2004), 273--292.

[66] M. J. Lai and P. Wenston, L1 Spline Methods for Scattered Data Interpolation and Approximation , Advances in Computational Mathematics 21(2004), 293--315.

[65] M.J.Lai and P. Wenston, Bivariate Splines for Fluid Flows, Computers and Fluids 33(2004), pp. 1047--1073.

[64] M. J. Lai, Chun Liu, and Paul Wenston, Bivariate spline method for numerical solution of time evolution Navier-Stokes equations over polygons in stream function formulation, Numerical Methods for P. D. E., 2003 pp. 776--827.

[63] W. He and M. J. Lai, Construction of trivariate compactly supported biorthogonal box wavelets, J. Approx. Theory, 120(2003), pp. 1--19.

[62] M. J. Lai and L. L. Schumaker, Macro-elements and stable local bases for splines on Powell-Sabin triangulations , Math. of Computation, 72(2003), 335--354.

[61] M. J. Lai, P. Wenston, L. A. Ying, Bivariate splines for exterior biharmonic equations, in Approximation Theory X: Wavelets, Splines and Applications, edited by C. K. Chui, L. L. Schumaker, J. Stoeckler, Vanderbilt Univ. Press, 2002, pp. 385--404.

[60] G. Awanou and M. J. Lai, $C^1$ quintic spline interpolation over tetrahedral partitions, in Approximation Theory X: Wavelets, Splines and Applications, edited by C. K. Chui, L. L. Schumaker, J. Stoeckler, Vanderbilt Univ. Press, 2002, pp. 1--16.

[59] M. J. Lai and D. Roach, Parameterizations of univariate orthogonal wavelets with short support, in Approximation Theory X: Wavelets, Splines, and Applications, edited by C. K. Chui, L. L. Schumaker, J. Stoeckler, Vanderbilt Univ. Press, 2002, pp. 369--384.

[58] M. J. Lai, Methods for Constructing Nonseparable Compactly Supported Orthonormal Wavelets, Wavelet Analysis: Twenty Year's Development, edited by D. X. Zhou, World Scientific, 2002, pp. 231--251.

[57] M. von Golitschek, M. J. Lai, L. L. Schumaker, Bounds for minimal energy bivariate polynomial splines, Numerische Math., 93(2002), 315--331.

[56] M. J. Lai and L. L. Schumaker, Quadrilateral macro-elements, SIAM J. Math. Anal., 33(2002), pp. 1107--1116.

[55] X. H. Wang, C. Li, and M. J. Lai, An Unified Convergence Theory for Newton's Type Methods for Zeros of Nonlinear Operators in Banach spaces , BIT, 42(2002), pp. 206--213.

[54] M. J. Lai and P. Wenston, Trivariate $C^1$ cubic splines for numerical solution of biharmonic equations, in: {\sl Trends in Approximation Theory}, K. Kopotun, T. Lyche, and M. Neamtu (eds.), Vanderbilt University Press, Nashville, 2001, pp. 224--234.

[53] M. J. Lai, and D. W. Roach, The nonexistence of bivariate symmetric wavelets with short support and two vanishing moments , in: {\sl Trends in Approximation Theory}, K. Kopotun, T. Lyche, and M. Neamtu (eds.), Vanderbilt University Press, Nashville, 2001, pp. 213--223.

[52] M. J. Lai and L. L. Schumaker, Macro-Elements and Stable Local Bases for Splines on Clough-Tocher Triangulations , Numerische Mathematik, 88(2001), pp. 105-119.

[51] M. J. Lai and P. Wenston, On Schwarz's domain decomposition methods for elliptic boundary value problems, Numerische Mathematik, 84(2000), pp. 475-495.

[50] M. J. Lai and P. Wenston, Bivariate spline method for numerical solution of Navier-Stokes equations over polygons in stream function formulation, Numerical Methods for P.D.E., 16(2000), 147--183.

[49] W. He and M. J. Lai, Examples of bivariate nonseparable compactly supported orthonormal continuous wavelets, IEEE Trans. Image Processing, 9(2000), 949--953.

[48] C. K. Chui and M. J. Lai, Filling polygonal holes using $C^1$ cubic triangular spline patches, Comp. Aided Geometric Design, 17(2000), 297--307.

[47] M. J. Lai, Convex preserving scattered data interpolation using bivariate $C^1$ cubic splines, J. Comput. Applied Math., 119(2000), 249--258.

[46] C. K. Chui, M. J. Lai, and J. Lian, Algorithms for $G\sp 1$ connection of multiple parametric bicubic NURBS surfaces , Numerical Algorithms, 23 (2000), 285--313.

[45] M. J. Lai and D. W. Roach, Nonseparable symmetric wavelets with short support, Proceedings of SPIE Conference on Wavelet Applications in Signal and Image Processing VII, Vol. 3813, pp. 132-146, July 1999.

[44] W. J. He and M. J. Lai, Construction of bivariate compactly supported biorthogonal box spline wavelets with arbitrarily high regularities, Applied Computational Harmonic Analysis, 6(1999), 53--74.

[43] M. J. Lai and L. L. Schumaker, On the approximation power of splines on triangulated quadrangulations, SIAM J. Numerical Analysis, 36(1999), pp. 143--159.

[42] M. J. Lai and P. Wenston, Bivariate Spline Method for Navier-Stokes Equations: Domain Decomposition Technique, in Approximation Theory IX: computational Aspects, C. K. Chui and L. L. Schumaker eds., Vanderbilt University Press, Nashville, 1998, pp. 153--160.

[41] K. Farmer and M. J. Lai, Scattered Data Interpolation by C^2 Quintic Splines Using Energy Minimization, in Approximation Theory IX: computational Aspects, C. K. Chui and L. L. Schumaker eds., Vanderbilt University Press, Nashville, 1998, pp. 47--54.

[40] W. J. He and M. J. Lai, A new sufficient condition for the orthonormality of refinable functions, in Approximation Theory IX: computational Aspects, C. K. Chui and L. L. Schumaker eds., Vanderbilt University Press, Nashville, 1998, pp. 121--128.

[39] W. J. He and M. J. Lai, Bivariate Box Spline Wavelets in Sobolev Spaces, in proceedings of SPIE, vol. 3458, 1998, pp. 56--66.

[38] M. J. Lai and L. L. Schumaker, Approximation power of bivariate splines, Advances in Comput. Math., 9(1998), pp. 251--279.

[37]W. J. He and M. J. Lai, Examples of bivariate nonseparable compactly supported orthonormal continuous wavelets, in Wavelet Applications in Signal and Image Processing IV, proceedings of SPIE, vol. 3169 (1997), pp. 303--314.

[36] W. J. He and M. J. Lai, On digital filters associated with bivariate box spline wavelets, Journal of Electronic Imaging, 6(1997), pp. 453--466.

[35] M. J. Lai and L. L. Schumaker, Scattered data interpolation using piecewise polynomials of degree six, SIAM Numer. Anal., 34(1997), pp.905--921.

[34] M. J. Lai, Geometric interpretation of smoothness conditions of triangular polynomial patches, Computer Aided Geometric Design, 14(1997), pp. 191-199.

[33] M. J. Lai, On $C^2$ quintic spline functions over triangulations of Powell-Sabin's type, Journal of Computational and Applied Mathematics, 73(1996), pp. 135--155.

[32] M. J. Lai, Bivariate box splines for image processing, in Wavelet Applications in Signal and Image Processing IV, proceedings of SPIE, vol. 2825 (1996), pp. 474--487.

[31] M. J. Lai and P. Wenston, On multilevel bases for elliptic boundary value problems, Journal of Computational and Applied Mathematics, 71(1996), pp. 95--113.

[30] M. J. Lai, On the fundamental solutions for multivariate singular interpolation, J. of Approx. Theory and Appl., 12(1996), 73--92.

[29] M. J. Lai, Scattered data interpolation and approximation by $C^1$ piecewise cubic polynomials, Computer Aided Geometric Design, 13(1996), pp. 81--88.

[28] M. J. Lai, Bivariate spline spaces on FVS-triangulations, in Approximation Theory VIII,, C. K. Chui and L. L. Schumaker, (eds.), Academic Press, 1995, pp. 309--316.

[27] M. J. Lai, On the digital filter associated with Daubechies' wavelets, IEEE Trans. Signal Processing, 43(1995), pp. 2203--2205.

[26] M. J. Lai, On computation of Battle-Lemarie's wavelets, Mathematics of Computation, 63(1994), pp. 689--699.

[25] M. J. Lai, On Str\"omberg's spline wavelets, Applied and Computational Harmonic Analysis, 1(1994), pp. 188-193.

[24] M. J. Lai, Approximation order from bivariate $C^1$ cubics on a four--directional mesh is full, Computer Aided Geometric Design, 11(1994), pp. 215--223.

[23] M. J. Lai, A matrix approach to computations of various wavelets, Proceedings of IMACS World Congress, 1(1994), pp. 284--286.

[22] M. J. Lai, Some Sufficient Conditions for Convexity of Multivariate Bernstein-Bezier Polynomials and Box Spline Surfaces, Studia Scientiarum Math. Hungarica, 28(1993), pp. 363--374.

[21] M. J. Lai, A Serendipity Family of Locally Supported Splines in S^2_8(\triangle), J. Approx. Theory and Appl. 10(1993), pp. 43--53.

[20] M. J. Lai, Asymptotic formulae of multivariate Bernstein approximation, J. Approx. Theory, 70(1992), pp229--242.

[19] M. J. Lai, A characteristic theorem of multivariate splines in blossom form, Computer Aided Geometric Design, 8(1992), pp513--521.

[18] C. K. Chui and M. J. Lai, Algorithms for generating B-nets and graphically displaying box spline surfaces, Computer Aided Geometric Design, 8(1992), pp479--493.

[17] M. J. Lai, Fortran subroutines for B-nets of box splines on three and four directional meshes, Numerical Algorithm, 2(1992), pp. 33--38.

[16] M. J. Lai, On dual functionals of polynomials in B-form, J. Approx. Theory, 67(1991), pp19--37.

[15] C. K. Chui and M. J. Lai, On bivariate super vertex splines, Constr. Approx., 6(1990), pp399-419.

[14] C. K. Chui and M. J. Lai, Multivariate vertex splines and finite elements, J. Approx. Theory, 60(1990), pp245-343.

[13] M. J. Lai, On construction of bivariate and trivariate vertex splines on mixed grid partitions, Ph. D. Dissertation, Texas A & M Unversity, 1989.

[12] M. J. Lai, A remark on integer translates of a box spline, J. Approx. Applic., 5(1989), pp97--104.

[11] G. Chen, C. K. Chui and M. J. Lai, Construction of real-time spline quasi-interpolation scheme, , J. Approx. Applic., 4(1988), pp 61--75.

[10] C. K. Chui and M. J. Lai, Multivariate analog of Marsden's identity and a quasi-interpolation scheme, Constr. Approx., 3 (1987), pp 111-122.

[9] C. K. Chui and M. J. Lai, On multivariate vertex splines and applications, in Topics in Multivariate Approximation, Chui, C.K., L.L. Schumaker, and F. Utreras eds. Academic Press, 1987, pp19-36.

[8] C. K. Chui and M. J. Lai, Computation of box splines and B-splines on triangulations of nonuniform rectangular partitions, J. Approx.Th. Applic., 3-4(1987), pp 37-62.

[7] C. K. Chui and M. J. Lai, VanderMonde determinants and Lagrange interpolation in {\bf R}^s, in Nonlinear and Convex analysis, B.L.Lin and S.Simons eds., Marcel Dekker, 1987, pp 23-35.

[6] M. J. Lai and X. H. Wang, On multivariate Newtonian interpolation, Scientia Sinica, 29 (1986), pp 23-32.

[5] C. K. Chui and M. J. Lai, On bivariate vertex splines, in Multivariate Approximation theory III, W.Schempp and K.Zeller, eds., Birkh\"auser, 1985, pp84-115.

[4] M. J. Lai, Exact error bounds for cubic Birkhoff spline interpolation, Numerical Math. J. Chinese Univ., 7(1985), pp. 369--372.

[3] G. Feng and M. J. Lai, On the uniform convergence of the Birkhoff interpolation with two points, Math. Numer. Sinica, 6(1984), pp 222-224.

[2] M. J. Lai and X. H. Wang, A note to the remainder of a multivariate interpolation polynomial, J. Approx. Appli., 1(1984), pp 57-63.

[1] M. J. Lai, On estimations for the exact bounds of the coefficients of approximation by cubic spline interpolation, Math. Numer. Sinica, 6(1984), pp 105-108.

Unpublished manuscripts

[2] C. K. Chui, M. J. Lai, and P. Wenston, Computation of Interpolatory Surfaces with Minimum Waviness, 2000.

[3] M. J. Lai and D. Roach, Construction of bivariate symmetric orthonormal wavelets with short support, 1999.

[4] W. He and M. J. Lai, Construction of bivariate nonseparable compactly supported orthonormal multiwavelets with arbitrarily high regularity, 1998.

Updated on May 20, 2006