Abstract
We prove that Kergin interpolation polynomials and Hakopian interpolation polynomials at the points of a Leja sequence for the unit disk D of a sufficiently smooth function f in a neighbourhood of D converge uniformly to f on D. Moreover, when f∈C ∞(D), all the derivatives of the interpolation polynomials converge uniformly to the corresponding derivatives of f.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. Andersson and M. Passare, Complex Kergin interpolation, J. Approx. Theory, 64 (1991), 214–225.
L. Białas-Cież and J.-P. Calvi, Pseudo Leja sequences, Ann. Mat. Pura Appl., available online doi:10.1007/s10231-010-0174-x.
T. Bloom, Kergin interpolation of entire function on ℂn, Duke Math. J., 48 (1981), 69–83.
T. Bloom and J.-P. Calvi, Kergin interpolants of holomorphic function, Constr. Approx., 13 (1997), 568–583.
T. Bloom and J.-P. Calvi, On distribution of extremal points for Kergin interpolantion: Real case, Ann. Inst. Fourier (Grenoble), 48 (1998), 205–222.
L. Bos and J.-P. Calvi, Kergin interpolant at the roots of unity approximate C 2 functions, J. d’Analyse Math., 72 (1997), 203–221.
J.-P. Calvi and V. M. Phung, Lagrange interpolation at real projections of Leja sequences for the unit disk, Proc. Amer. Math. Soc. (accepted).
J.-P. Calvi and V. M. Phung, On the Lebesgue constant of Leja sequences for the disk and its applications to multivariate interpolation, J. Approx. Theory, 163 (2011), 608–622.
L. Filipsson, Complex mean-value interpolation and approximation of holomorphic function, J. Approx. Theory, 91 (1997), 244–278.
T. N. T. Goodman, Interpolation in minimum seminorm and multivariate B-spline, J. Approx. Theory, 37 (1983), 212–223.
H. A. Hakopian, Multivatiate divided differences and multivariate interpolation of Lagrange and Hermite type, J. Approx. Theory, 34 (1982), 286–305.
X. Z. Liang, On Hakopian interpolation in the disk, J. Approx. Theory Appl., 2 (1986), 37–45.
X. Z. Liang, R. Feng and X. Sun, Weighted mean convergence of Hakopian interpolation on the disk, Anal. Theory Appl., 23 (2007), 213–227.
X. Z. Liang and C. M. Lü, On the convergence of Hakopian interpolation and cubature, J. Approx. Theory, 88 (1997), 28–46.
C. A. Micchelli, A constructive approach to Kergin interpolation in R k: multivariate B-spline and Lagrange interpolation, Rocky Mountain J. Math., 10 (1980), 485–497.
D. L. Ragozin, Constructive polynomial approximation on spheres and projective spaces, Trans. Amer. Math. Soc., 162 (1971), 157–170.
Y. Sarantopoulos, Bounds on the derivatives of polynomials on Banach spaces, Math. Proc. Cambridge Philos. Soc., 110 (1991), 307–312.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Phung, V.M. On the convergence of Kergin and Hakopian interpolants at Leja sequences for the disk. Acta Math Hung 136, 165–188 (2012). https://doi.org/10.1007/s10474-012-0239-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10474-012-0239-y