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Bivariate Lagrange interpolation based on Chebyshev points of the second kind

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Abstract

We give compact formulae of the Lagrange interpolation polynomials and cubature formulae based on the common zeros of product Chebyshev polynomials of the second kind. Further, for \(0<p\leq2\), we study the mean convergence of the Lagrange interpolation polynomials.

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References

  1. Bojanov, B., Petrova, G.: On minimal cubature formulae for product weight functions. J. Comput. Appl. Math. 85, 113–121 (1997)

    Article  MathSciNet  Google Scholar 

  2. P. Borwein and T. Erdélyi, Polynomials and Polynomial Inequalities, Springer-Verlag (New York, 1992)

  3. Dunkl, C.F., Xu, Y.: Orthogonal Polynomials of Several Variables. Cambridge University Press (2003)

  4. Lubinsky, D.S., Máté, A., Nevai, P.: Quadrature sums involving \(p\)th powers of polynomials. SIAM J. Math. Anal. 18, 531–544 (1987)

    Article  MathSciNet  Google Scholar 

  5. Möller, H.M.: Kubaturformeln mit minimaler Knotenzahl. Numer. Math. 25, 185–200 (1976)

    Article  MathSciNet  Google Scholar 

  6. Morrow, C.R., Patterson, T.N.L.: Construction of algebraic cubature rules using polynomial ideal theory. SIAM J. Numer. Anal. 15, 953–976 (1978)

    Article  MathSciNet  Google Scholar 

  7. Mysovskikh, I.P.: On the construction of cubature formulas with fewest nodes. Soviet Math. Dokl. 9, 277–280 (1968)

    MATH  Google Scholar 

  8. Radon, J.: Zur mechanischen Kubatur. Monatsh. Math. 52, 286–300 (1948)

    Article  MathSciNet  Google Scholar 

  9. Schmid, H.J.: On cubature formulae with a minimal number of knots. Numer. Math. 31, 281–297 (1978)

    Article  MathSciNet  Google Scholar 

  10. Stroud, A.H.: Approximate Calculation of Multiple Integrals, Prentice Hall. Englewood Cliffs, NJ (1971)

    MATH  Google Scholar 

  11. J. Szabados and P. Vértesi, Interpolation of Functions, World Scientific (Singapore, 1990)

  12. Xu, Y.: Gaussian cubature and bivariate polynomial interpolation. Math. Comp. 59, 547–555 (1992)

    Article  MathSciNet  Google Scholar 

  13. Xu, Y.: On multivariate orthogonal polynomials. SIAM J. Math. Anal. 24, 783–794 (1993)

    Article  MathSciNet  Google Scholar 

  14. Xu, Y.: Common Zeros of Polynomials in Several Variables and Higher Dimensional Quadrature, Pitman Research Notes in Mathematics Series 312, Longman Scientific & Technical. Essex, U.K. (1994)

    Google Scholar 

  15. Xu, Y.: Christoffel functions and Fourier series for multivariate orthogonal polynomials. J. Approx. Theory 82, 205–239 (1995)

    Article  MathSciNet  Google Scholar 

  16. Xu, Y.: Lagrange interpolation on Chebyshev points of two variables. J. Approx. Theory 87, 220–238 (1996)

    Article  MathSciNet  Google Scholar 

  17. A. Zygmund, Trigonometric Series, Vols. I, II, 2nd ed., Cambridge Univ. Press (Cambridge, 1977)

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Acknowledgement

We express our gratitude to the referee for helpful suggestions.

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Authors and Affiliations

  1. School of Mathematics, Renmin University of China, Beijing, 100872, China

    J. Liu & L. Y. Zhu

  2. School of Mathematics and Physics, Handan University, Handan, 056005, China

    J. Liu

Authors
  1. J. Liu
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  2. L. Y. Zhu
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Corresponding author

Correspondence to J. Liu.

Additional information

Supported by the Science and Technology Research Project of University of Hebei Province (Grant No. ZC2016066), the Research and Development Program for Science and Technology of Handan (Grant No. 1523103064-6) and the Nursery Engineering of Handan University (Grant No. 2018303).

Laiyi Zhu is supported by the National Natural Scince Foundation of China (Grant No. 11571362).

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Liu, J., Zhu, L.Y. Bivariate Lagrange interpolation based on Chebyshev points of the second kind. Acta Math. Hungar. 159, 618–637 (2019). https://doi.org/10.1007/s10474-019-00988-5

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  • DOI: https://doi.org/10.1007/s10474-019-00988-5

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