Abstract
Classical rational interpolation usually enjoys better approximation properties than polynomial interpolation because it avoids wild oscillations and exhibits exponential convergence for approximating analytic functions. We develop a rational interpolation operator, which not only preserves the advantage of classical rational interpolation, but also has a finite Lebesgue constant. In particular, it is convergent for approximating any continuous function, and the convergence rate of the interpolants approximating a function is obtained using the modulus of continuity.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Berrut, J.-P., Elefante, G.: A periodic map for linear barycentric rational trigonometric interpolation. Appl. Math. Comput. 371, 124924 (2020)
Berrut, J.-P., Klein, G.: Recent advances in linear barycentric rational interpolation. J. Comput. Appl. Math. 259, 95–107 (2004)
Berrut, J.-P.: Rational functions for guaranteed and experimentally well-conditioned global interpolation. Comput. Math. Appl. 15(1), 1–16 (1988)
Berrut, J.-P., Trefethen, L.N.: Barycentric Lagrange interpolation. SIAM Rev. 46, 501–517 (2004)
Bos, L., De Marchi, S., Hormann, K.: On the Lebesgue constant of Berrut’s rational interpolant at equidistant nodes. J. Comput. Appl. Math. 236(4), 504–510 (2011)
Bos, L., De Marchi, S., Hormann, K., Sidon, J.: Bounding the Lebesgue constant for Berrut’s rational interpolant at general nodes. J. Approx. Theory 169, 7–22 (2013)
Bos, L., De Marchi, S., Hormann, K., Klein, G.: On the Lebesgue constant of barycentric rational interpolation at equidistant nodes. Numer. Math. 121(3), 461–471 (2012)
Brutman, L.: Lebesgue functions for polynomial interpolation—a survey. Ann. Numer. Math. 4, 111–127 (1997)
Carnicer, J.M.: Weighted interpolation for equidistant nodes. Numer. Algorithms 55(2–3), 223–232 (2010)
Cheney, W., Light, W.: A Course in Approximation Theory. American Mathematical Society, Providence (2000)
Floater, M.S., Hormann, K.: Barycentric rational interpolation with no poles and high rates of approximation. Numer. Math. 107(2), 315–331 (2007)
Smith, S.J.: Lebesgue constants in polynomial interpolation. Ann. Math. Inform. 33, 109–123 (2006)
Trefethen, L.N., Weideman, J.A.C.: Two results on polynomial interpolation in equally spaced points. J. Approx. Theory 65(3), 247–260 (1991)
Wang, Q., Moin, P., Iaccarino, G.: A rational interpolation scheme with superpolynomial rate of convergence. SIAM J. Numer. Anal. 47(6), 4073–4097 (2010)
Zhang, R.: Optimal asymptotic Lebesgue constant of Berruts rational interpolation operator for equidistant nodes. Appl. Math. Comput. 294, 139–145
Zhang, R.: An improved upper bound on the Lebesgue constant of Berrut’s rational interpolation operator. J. Comput. Appl. Math. 255(1), 652–660 (2014)
Acknowledgements
This work is supported by the National Natural Science Foundation of China (Grant No. 61772025) and Natural Science Foundation of Zhejiang Province (No. LY20F020004).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Zhang, RJ., Liu, X. Rational interpolation operator with finite Lebesgue constant. Calcolo 59, 10 (2022). https://doi.org/10.1007/s10092-021-00454-1
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10092-021-00454-1