Publications:

                                                                 

           84.    Li Jian, Liu Yongping,   The Jackson inequality  for the best L^2-Approximation of functions on [0, 1] with the weight x, Numerical Mathematics:Theory, Methods and Applications, 1:3(2008), 340-356.          

     

           83.  Li Rensuo, Liu Yongping,  the Asymptotic estimations of best m-term one-side approximation of function classes determined by Fourier series,  Advance in Mathematics (CHINA), 37:2(2008), 211-221.

          

           82. Liu Yongping Xiao Weiwei, Relative average widths of Sobolev spaces in L 2(R d ) , Anaysis Mathematica, 34(2008), 71-82.

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           81. Liu Yongping, Yang Lianhong,  Relative widths of smooth functions determined by fractional order derivatives, J. Complexity, 24:2(2008), 259-282.
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          80.  Li Rensuo, Liu Yongping, Asymptotic estimations of m-term approximation and greedy algorithm for multiplier function classes defined by Fourier series,  Chinese J. Engineer Mathematics, 25:1(2008), 89-96.

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          79.    Yang Zhuyuan, Yang Zongwen,  Liu Yongping,  Average onesided widths of anisitropic Soblev and Besov classess, Acta Mathematca Sinica, Chinese Series,  50:5(2007), 1177-1184.        

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         78.     Gui Shaohui,  Liu Yongping, Asymptotic estimate for the Lesque constant of cardinal L-spline interpolation operator, East Journal on Approximation, 3(2007), 331-355.     

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         77.   Hui Cao, Yongping Liu.  Lebesgue constants on Gaussian cardinal interpolation operators from l(Z) into L(R). Analysis Mathematica, 32(2006),  265-277.

 

     76.  Liu Yongping, Yang Lianhong,  Relative width of smooth classes of multivariate periodic functions with restrictions on iterated Laplace derivatives in the L_2-metric,  Acta Math. Scientia, 26B:4, (2006), 720-728.  

   

         75. Gui Qiao XU,  Yong Sheng  SUN,  Yong Ping LIU.  N- widths  and  average widths of Besov classes in Sobolev spaces, Acta Math. Sinica, English Series,  22:6 (2006),  1667-1678.  

          

         74.   Yang Zhuyuan, Liu Yongping, Approximation by cardinal  Hermite splines, J.  Yunan University, 28:5(2006), 374-377.

          

         73.   Yang Zhuyuan, Liu Yongping, Approximation of Lipschitz spaces from cardinal splines, J. Yunan University, 28:4(2006), 281-284.

          

         72.   Yang Zhuyuan, Liu Yongping, Approximation of Multivariate Cardinal Splines, J. Beijing Normal University, 42:3(2006), 221-223.

          

         71.  Gui Shaohui,  Liu Yongping, Strong asymptotic estimate for Lesque constant on Gaussian cardinal interpolation operator, J. Beijing Normal University, 42:2(2006), 116-119.

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         70.  Hu Yingkang, Liu Yongping,  On equivalence of moduli of smoothness of polynomials in $L\sb p$, $0<p \leq \infty$.  J. Approx. Theory, 136:2(2005), 182--197.

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         69.  Zhou Heng, Liu Yongping, Shi Limin, On the convergence of a positive linear operator sqeuence,  J.  Beijing  Normal University, 41:4(2005), 346-347.

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         68.  Li Rensuo, Liu Yongping, Asymptotic estimate of m-term approximation and greedy algorithm of the classes defined by a multiplier, J. Beijing Normal University, 41:2(2005), 127-130.

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        67.  Yang Lianhong, Liu Yongping, Relative widths of classes of multivariate differentiable functions in the space L_2(R^d), J. Beijing Normal University, 41:1(2005), 21-23.

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        66.  Ren Meiying, Liu Yongping. A 2-periodic entire (m_1,m_2; m'_1, m'_2) interpolation problem on equidistant nodes,  Journal of Tianjin Normal University,  24:3 (2004), 37-41.   

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    65.  Yongping  Liu;  Guiqiao Xu. Infinite-dimensional Gel'fand \sigma widths of anisotropic Besov classes,  PanAmerican Mathematical Journal,  14:3(2004), 65-77.

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    64.  Xu, Guiqiao; Liu, Yongping. The  average widths of Sobolev-Wiener classes and Besov-Wiener classes, Acta Math. Sinica, English Series,  20:1 (2004), 81-92.  

        

    63.  Liu, Yongping; Xu, Guiqiao. Widths and average widths of Sobolev classes.  Acta Math. Scientia, 23B:2, (2003), 178-184.  SCI 

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    62. Liu, Yongping; Xu, Guiqiao.  The Average widths and non-linear  widths of the classes  of multivariate functions with bounded moduli of smoothness.  Acta Math. Applicatae Scientia (English Series), 18:4, (2002), 663-674. 

    

       61.  Xu, Guiqiao; Liu, Yongping. The Gel' fand widths of anisotropic Sobolev classes of smooth functions. Anal. Math. 28 (2002), no. 4, 317--332.

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      60.  Liu, Yongping; Xu, Guiqiao. The infinite-dimensional widths and optimal recovery of generalized Besov classes.  J. Complexity 18 (2002), no. 3, 815--832.   SCI

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   59.  Liu, Yongping;  Lu, Guozhen; Wheeden, Richard L..  Some equivalent definitions of high order Sobolev spaces on stratified groups and generalizations to metric spaces.  Math. Ann. 323 (2002), no. 1, 157--174. SCI
      

   58.  Jiang, Yanjie and  Liu, Yongping.  Average widths and optimal recovery of multivariate Besov classes in the amalgams of  Lq  and  lp .    J. of Computational Analysis and Applications,  3:4(2001), 301-306.   SCI

   

   57. Yang, Zhuyuan and Liu, Yongping.  Approximation by cardinal splines. (Chinese) Beijing Shifan Daxue Xuebao, 37 :5(2001), no. 5, 592--595.

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   56.  Xu Guiqiao and   Liu Yongping .  The rate of  L_p convergence of the Gr\"unwald process.  J. Math. Research   and  Exposition, 21:3(2001), 447-451.   

55.  Liu Yongping  and  Xu Guiqiao, Some extrimal properties of multivariate cardinal splines in the metric  L_p(R^d), Zhongguo Kexue (Series A), 31:4(2001), 307-314 (in Chinese); Science in China(Series A), 44:8(2001), 961-968. SCI 

54.  Kang Shugui   and   Liu Yongping , The problem of A. F. Timan On the precise order of the best approximations of multivariate functions, J. Math. Research   and  Exposition, 21:2(2001), 177-184. 

53.   Xu Guiqiao   and   Liu Yongping , The Infinite-dimensional 考-widths of Besov classes, Advances in Mathematics , 30:1(2001), 89-90. (Research Announcements).   

52.  Liu Yongping ,  L2(Rn)- Optimal recovery on the Riesz potential space, J. Math. Research and Exposition,  20:3(2000), 365-378. 

51.  Jiang Yanjie and  Liu Yongping , Average widths and optimal recovery of some classes of multivariate smooth functions, Approximation Theory of Functions and its Applications, Ukraine 31, 2000, 215-226. 

50.  Xu Guiqiao  and  Liu Yongping , The Gel'fand  n- widths of Besov classes of multivariate periodic smoothness functions, J. Beijing Normal University (Natural Science), 36:6(2000), 748-751. 

49.  Xu Guiqiao and Liu Yongping , Some extreme properties of multivariate cardinal splines, J. Beijing Normal University (Natural Science),  36:5(2000), 607-611.    

48.  Xu Guiqiao  and  Liu Yongping ,The rate of weighted  L_2- converge rate of a kind of quasi-Gr\"unwald interpolatory operators,  J. of Engineering Math.,  17:2(2000), 20-24.  

47.  Kang Shugui and Liu Yongping , On the precise order of the best approximations of multivariate functions, J. Math. Study, 33:1(2000), 1-8.  

46.  Jiang Yanjie and Liu Yongping , Average widths and optimal recovery of multivariate Besov classes in  L_p(R^d) , J. Approx. Theory  102, (2000), 155-170.  SCI

45.  Yongping Liu  and  Guozhen Lu, Approximation properties of the derivatives of polyharmonic interpolation splines in  L_p(R^n) space,     J. Approx. Theory 101,  (1999), 49-62.  SCI

44.  Sun Yongsheng,  Liu Yongping    and  Chen Dirong, Extreme problems in approximation theory for some classes of smooth functions defined on  R^d ,    J. Beijing Normal University (Natural Science), Vol.35(Sup.), (1999), 79-144.

 43.  Luo Junbo   and   Liu Yongping , Average width and optimal recovery of some anisotropic classes of smooth functions defined on the Euclidean space  R^d ,     J. Northeast Math.,  15:4(1999), 423-432.  

42.  Liu Yongping and  Xu Guiqiao, The rate of weight  L_2- convergence of the Grunwald interpolation operators, J. of Engineering Math., 16:4 (1999), 104-106.    

41.   Yongping Liu , Guozhen Lu   and  Richard L. Wheeden, Representation formulas and Sobolev spaces of higher order on stratified groups and generalizations to metric spaces,    Math. Sci. Res. Hot-line, 3(7)(1999), 35-59.  

40.  Wang Meiying and  Liu Yongping , Convergence rates  for lacunary trigonometric interpolation in  \tilde L _ 2\pi ^p , Approx.Theory and its Appl.,  15:2(1999), 90-99.  

39.  Jiang Yanjie and Liu Yongping, Average  B-width and infinite dimensional  G- width of some smooth function classes on the line, Approx. and  its Appl., 15:1 (1999), 50-63.  

38.  Liu Yongping , Approximation of smooth functions by polyharmonic cardinal splines in  Lp(Rn) space,  Acta Mathematicae Applicatae Sinica,  14:2 (1998), 157-164.  

37.  Liu Yongping , Infinite dimensional widths of function classes with non-symmetric smoothness conditions, in  Approximation Theory IX, Vol.I,(Nashville, TN, 1998), 231-238, Charles K. Chi and Larry L. Schumaker (eds.), Innov. Appl. Math.,Vanderbilt Univ. Press, Nashville, YN. 1998. 

36.  Jiang Yanjie   and   Liu Yongping , Average widths and optimal recovery of multivariate Besov-Wiener classes, J. Math. Study,  31:4(1998), 353-361.  

35.  Luo Junbo and Liu Yongping , Average width and optimal recovery of the anisotropic classes  W^rH^\alpha(R^d)  in the metric  C^k(R^d) , Analysis Mathematica,  24(1998), 265-273.  

34.  Yang Zhuyuan  and  Liu Yongping , Average onesided widths of some classes of smooth functions defined on  R^d ,  Advances in Mathematics,  27:3(1998), 278-280.  

33.  Jiang Yanjie   and   Liu Yongping , Average widths and optimal recovery of multivariate Besov classes in  L_p(R^d) ,  Chinese Science Bulletin (Kexue Tongbao),  43:8(1998), 638-641;  Kexue Tongbao,  (1998), 479-482 (in Chinese).  SCI

32.  Jiang Yanjie  and  Liu Yongping , Average  B- width and infinite dimensional  G- width of some smooth function classes on the line,  Chinese Science Bulletin (Kexue Tongbao),  43:10(1998), 810-812;   Kexue Tongbao,  (1998), 701-703 (in Chinese). SCI  

31.  Liu Yongping ,  Lq(Rm)- Optimal recovery on the Riesz potential spaces with imcomplete information,  J. Beijing Normal University (Natural Science),  33:2(1997), 143-150. 

30.  Yang Zhuyuan   and   Liu Yongping , Approximation from elliptic-cardinal spline,    J. Beijing Normal University (Natural Science),  33:4(1997), 440-443.(In Chinese).    

29.  Fang Gensun  and  Liu Yongping , Optimal quadrature of  W^r_1(R)  on whole real axis,  Acta Mathematica Scientia, 16:1(1996), 72-80.    

28.  Liu Yongping, Multivariate approximation on the basis of information containing errors,  East Journal on Approximations,  2:4(1996), 463-476.    

27.  Liu Yongping, L2(Rn)- extreme problems on the basis of incomplete information, Analysis Mathematica, 21(1995), 107-124.    

26.  Liu Yongping , Average  考-K  width of convolution function class of  L(R)  in  L2(R) ,  J. Math.Research   and  Exposition,  15:2(1995), 199-207.  

25.   Liu Yongping , Average 考-K  width of class of smooth functions of  Lp(Rn)   in  Lq(Rn), Chin. Ann. of Math., 16B:3(1995), 351-360.  

24.  Liu Yongping , Optimal recovery of class of smooth functions defined on the whole real axis by multifold sampling,  Acta Mathematica Sinica (new series)  1995, Vol.11, Special Issue (July), 128-141.  

23.  Liu Yongping, Average  考-K  width of convolution function class of  L pq (R) in Lq(R), Approx. Theory  and  its Appl.  10:1 (1994), 34-46.  

22.  Liu Yongping , Optimal recovery of class of smooth functions defined on the whole real axis by multifold sampling, Kexue Tongbao,  38:14 (1993), 1266-1270 (in  Chinese); also see  Chinese Science Bulletin, 38:20 (1993), 1683-1687. 

21.  Liu Yongping , Infinite dimensional widths and optimal recovery of some  smooth function classes of  Lp(R)  in metric  L(R) ,  Kexue Tongbao, 38:18 (1993), 1642-1645(in Chinese); also see  Chinese Science Bulletin, 38:20 (1993), 1678-1682.    

20.  Liu Yongping , Infinite dimensional widths and optimal recovery of some smooth function classes of  Lp(R)  in metric  L(R) , Analysis Mathematica, 19:2 (1993), 169-182.    

19.  Liu Yongping , Average 考-K  width of class of smooth functions with bounded modulus of continuity in  L2(R) , J. Beijing Normal University (Natural Science),  29:1 (1993), 44-49 (in Chinese). 

18.  Liu Yongping , Optimal quadrature formulas on some classes of smooth functions on the whole real axis, Northeastern Math. J.,  9:1 (1993), 113-120. 

17.  Fang Gensun   and   Liu Yongping , Optimal quadrature of  W^r_1(R)  on whole real axis and minimal radius of information, J. Beijing Normal  University (Natural Science), 29:4 (1993), 454-457. 

16.  Fang Gensun   and   Liu Yongping , Average width and optimal interpolation of the Sobolev-Wiener class  W^r_ pq (R)  in the metric  L_q(R)  ,    J. Approx. Theory,   74:3 (1993), 335-352.  SCI

15.  Sun Yongsheng   and   Liu Yongping , Optimal recovery of Sobolev-Wiener class of smooth functions by double sampling,   Constr. Approx.,  Vol.9, (1993), 391-405.  SCI

14.  Hou Xiangqian   and   Liu Yongping , Equiconvergence of some trigonometric interpolation polynomials,  J. Beijing Normal University (Natural Science),  Vol.28, Sup.2  (1992), 15-20.  

13.  Sun Yongsheng   and   Liu Yongping ,  N 每widths for some classes of periodic functions with boundary conditions,    Approx. Theory   and  its Appl.,  8:3 (1992), 21-27.  

12.  Liu Yongping and  Sun Yongsheng, Infinite-dimensional   Kolmogorov width and optimal interpolation on Sobolev-Wiener space,  Scientia Sinica (ser. A),  6 (1992), 582-591; also see  Sinence in China (ser. A),  35:10 (1992), 1162-1172.   SCI

     11.  Fang Gensun and  Liu Yongping , Average width and  optimal interpolation of the Sobolev-Wiener class  W^r_ pq (R)  in the metric L_q(R),  J. Beijing Normal University (Natural Science), 28:3 (1992), 306-309. 

      10.  Sun Yongsheng and  Liu Yongping , Optimal recovery of Sobolev-Wiener class of smooth functions by double sampling, J. Beijing Normal University (Natural Science),  28:1 (1992), 15-18.    

9.   Fang Gensun and  Liu Yongping , On the east  L_\infty -norm of monosplines with free knots on the real axis,    J. Complexity,  8 (1992), 467-487.   SCI

8.  Liu Yongping , Average  n-K  width of some generalized Sobolev-Wiener classes, Approx. Theory   and  its Appl., 8:4 (1992), 79-88.  

7.  Liu Yongping , Approximation of signed measures and functions on the cube  T_n  by  B-R  spherical means on the total measure set,    Chinese Ann. Math.,  13A:5 (1992), 610-621 (in Chinese); also see  Chinese J. Contemporary Math.,  13:4 (1992), 341-355. 

6.  Liu Yongping, Approximation properties of certain interpolation operators of entire exponential type in Lp(-﹢,+﹢) spaces,  Acta Mathematica Sinica, New Series, 7:4 (1991), 289--308. 

5.  Liu Yongping, On the entire (0, m_1,＃, m_q)-interpolation on equidistant nodes, Approx. Theory and its Appl., 7:3 (1991), 27--34. 

4.  Sun Yongsheng  and  Liu Yongping , Best one-side approximation of some classes of smooth functions on the whole real axis by cardinal splines of higher order,    Approx. Theory and its Appl., 7:1 (1991), 108-123. 

3.  Liu Yongping, On the trigonometric interpolation and the entire interpolation,  Approx. Theory and its Appl., 6:4 (1990), 85--106. 

2.  Liu Yongping, The relations of two moduli of continuity for functions on (-﹢,+﹢),  J. Beijing Normal University (Natural Science), No.4,   (1989), 1-6. (In Chinese). 

1.  Liu Yongping, The approximation properties of certain trigonometric interpolation polynomial operators, Northeastern Math. J., 4 (1988), 289-308. (In Chinese).