Abstract
In his paper ``Lagrange interpolation on Chebyshev points of two variables'' (J. Approx. Theor. 87, 220–238, 1996), Y. Xu proposed a set of Chebyshev like points for polynomial interpolation in the square [−1,1]2, and derived a compact form of the corresponding Lagrange interpolation formula. We investigate computational aspects of the Xu polynomial interpolation formula like numerical stability and efficiency, the behavior of the Lebesgue constant, and its application to the reconstruction of various test functions.
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L. Bos (1991) ArticleTitleOn certain configurations of points in ℝn which are unisolvent for polynomial interpolation J. Approx. Theor. 64 IssueID3 271–280 Occurrence Handle0737.41002 Occurrence Handle1094439
L. Brutman (1997) ArticleTitleLebesgue functions for polynomial interpolation—a survey Ann. Numer. Math 4 111–127 Occurrence Handle0888.41001 Occurrence Handle1422674
M. Caliari S. Marchi ParticleDe M. Vianello (2005) ArticleTitleBivariate polynomial interpolation on the square at new nodal sets Appl. Math. Comput. 165 261–274 Occurrence Handle10.1016/j.amc.2004.07.001 Occurrence Handle2137790 Occurrence Handle1081.41001
P. J. Davis (1975) Interpolation and approximation Dover Publications New York Occurrence Handle0329.41010
M. Dubiner (1995) ArticleTitleThe theory of multi-dimensional polynomial approximation J. Anal. Math. 67 39–116 Occurrence Handle0857.41006 Occurrence Handle1383490 Occurrence Handle10.1007/BF02787786
M. Gasca T. Sauer (2000) ArticleTitlePolynomial interpolation in several variables Adv. Comput. Math. 12 377–410 Occurrence Handle10.1023/A:1018981505752 Occurrence Handle1768957 Occurrence Handle0943.41001
C. R. Morrow T. N. L. Patterson (1978) ArticleTitleConstruction of algebraic cubature rules using polynomial ideal theory SIAM J. Numer. Anal. 15 IssueID5 953–976 Occurrence Handle10.1137/0715062 Occurrence Handle507557 Occurrence Handle0402.65013
W. Pleśniak (1996) ArticleTitleRemarks on Jackson's theorem in ℝN East J. Approx. 2 IssueID3 301–308 Occurrence Handle1413170 Occurrence Handle0860.41014
L. Reichel (1990) ArticleTitleNewton interpolation at Leja points BIT 30 332–346 Occurrence Handle10.1007/BF02017352 Occurrence Handle0702.65012 Occurrence Handle1039671
T. Sauer (1995) ArticleTitleComputational aspects of multivariate polynomial interpolation Adv. Comput. Math. 3 219–238 Occurrence Handle0831.65006 Occurrence Handle1325032
T. Sauer Y. Xu (1995) ArticleTitleOn multivariate Lagrange interpolation Math. Comp. 64 1147–1170 Occurrence Handle1297477 Occurrence Handle0823.41002
H. Tal-Ezer (1991) ArticleTitleHigh degree polynomial interpolation in Newton form SIAM J. Sci. Stat. Comput. 12 IssueID3 648–667 Occurrence Handle10.1137/0912034 Occurrence Handle0728.65009 Occurrence Handle1093210
Tang, P. T. P.: Some software implementations of the functions sine and cosine. ANL Report 90/3, Argonne National Laboratory, April 1990.
Y. Xu (1992) ArticleTitleGaussian cubature and bivariate polynomial interpolation Math. Comp. 59 547–555 Occurrence Handle0766.65006 Occurrence Handle1140649
Y. Xu (1996) ArticleTitleLagrange interpolation on Chebyshev points of two variables J. Approx. Theor. 87 220–238 Occurrence Handle0864.41002 Occurrence Handle10.1006/jath.1996.0102
Y. Xu (2004) ArticleTitlePolynomial interpolation on the unit sphere and on the unit ball Adv. Comput. Math. 20 247–260 Occurrence Handle10.1023/A:1025851005416 Occurrence Handle1047.41006 Occurrence Handle2055799
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Bos, L., Caliari, M., De Marchi, S. et al. A Numerical Study of the Xu Polynomial Interpolation Formula in Two Variables. Computing 76, 311–324 (2006). https://doi.org/10.1007/s00607-005-0137-z
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DOI: https://doi.org/10.1007/s00607-005-0137-z