JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2016, Vol. 51 ›› Issue (10): 11-15.doi: 10.6040/j.issn.1671-9352.0.2015.541

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The average error of linear tensor product multivariate polynomial interpolation based on Chebyshev nodes on the Brownian sheet measure

XIONG Li-yan, XU Gui-qiao*   

  1. College of Mathematical Science, Tianjin Normal University, Tianjin 300387, China
  • Received:2015-11-18 Online:2016-10-20 Published:2016-10-17

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Abstract: Based on univariate Lagrange polynomial interpolation, a kind of linear tensor product polynomial interpolation is constructed to approximate multivariate functions. For the weighted L2-norm,their average errors is studied on the Brownian sheet measure and obtained the corresponding stronger asymptotic order. Compared with the past algorithms based on linear functional information, our algorithms are based on standard information and it is constructive. It can also be applied to solve practical problems. From the perspective of average error, it is showed that algorithms are order optimal to the univariate function case setting, and have a similar approximation order to the optimal algorithms based on linear functional information to the multivariate function case setting.

Key words: Chebyshev nodes, average error, weighted L2-norm, Brownian sheet measure

CLC Number: 

  • O174.41
[1] TRAUB J F, WASILKOWSKI G W, WOZNIAKOWSKI H. Information-Based Complexity[M]. New York: Academic Press, 1988.
[2] KLAUS R. Average-case analysis of numerical problems[M]. New York: Springer-Verlag, 2000.
[3] SULDIN A V. Wiener measure and its applications to approximation methods I[J]. Izv Vyssh Ucheb Zaveb Mat, 1959, 13:145-158.
[4] SULDIN A V. Wiener measure and its applications to approximation methods II[J]. Izv Vyssh Ucheb Zaved Mat, 1960, 18:165-179.
[5] NOVAK E, WOZNIAKOWSKI H. Tractability of multivariate problems, standard information for operator[M]. Switzerland:European Mathematical Society, 2012.
[6] 许贵桥. Lagrange插值和Hermite-Fejér插值在Wiener空间下的平均误差[J]. 数学学报, 2007, 50(6):1281-1296. XU Guiqiao. The average error for lagrange interpolation and Hermite-Fejér interpolation on the wiener space[J]. Acta Mathematica Sinica, 2007, 50(6):1281-1296.
[7] LIFSHITS M. Lectures on Gaussian processes[M]. New York: Springer, 2012.
[8] VARMA A K, PRASAD J. An analogue of a problem of P. Erdös and E. Feldheim on Lp convergence of interpolatory processes[J]. Journal of Approximation Theory, 1989, 56(2):225-240.
[9] WASILKOWSKI G W. Integration and approximation of multivariate functions: average case complexity with isotropic Wiener measure[J]. Bull Amer Math Soc(N. S.), 1993, 28:308-314.
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