Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Spline interpolation at knot averages on a two-sided geometric mesh


Author: M. J. Marsden
Journal: Math. Comp. 38 (1982), 113-126
MSC: Primary 41A15; Secondary 65D07
DOI: https://doi.org/10.1090/S0025-5718-1982-0637290-3
MathSciNet review: 637290
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For splines of degree $ k \geqslant 1$ with knots $ - {t_i} = {t_{2m + 1 - i}} = 1 + q + {q^2} + \cdots + {q^{m - i}}$, $ i = 1, \ldots ,m$, where $ 0 < q < 1$, it is shown that spline interpolation to continuous functions at nodes $ {\tau _i} = \Sigma _1^k{w_j}{t_{i + j}}$, $ i = 1, \ldots ,n = 2m - k - 1$, has operator norm $ \left\Vert P \right\Vert$ which is bounded independently of q and m as q tends to zero if and only if $ {(1 - {w_1})^k} < \frac{1}{2}$, $ {(1 - {w_k})^k} < \frac{1}{2}$, and $ {w_j} > 0$, $ j = 1, \ldots ,k$. The choice of nodes $ {\tau _i} = \Sigma _0^{k + 1}{w_j}{t_{i + j}}$ and the growth rate of $ \left\Vert P \right\Vert$ with k are also discussed.


References [Enhancements On Off] (What's this?)

  • [1] C. de Boor, A Practical Guide to Splines, Applied Mathematical Sciences Series, Vol. 27, Springer-Verlag, New York, 1978. MR 507062 (80a:65027)
  • [2] C. de Boor, ``On bounding spline interpolation,'' J. Approx. Theory, v. 14, 1975, pp. 191-203. MR 0382911 (52:3793)
  • [3] S. Demko, ``Interpolation by quadratic splines,'' J. Approx. Theory, v. 23, 1978, pp. 392-400. MR 509568 (80e:41006)
  • [4] H. W. Gould, Combinatorial Identities, Henry W. Gould, Morgantown, 1972. MR 0354401 (50:6879)
  • [5] S. L. Lee, private communication.
  • [6] F. Richards, ``The Lebesgue constants for cardinal spline interpolation,'' J. Approx. Theory, v. 14, 1975, pp. 83-92. MR 0385391 (52:6254)
  • [7] T. J. Rivlin, An Introduction to the Approximation of Functions, Blaisdell, New York, 1969. MR 0249885 (40:3126)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 41A15, 65D07

Retrieve articles in all journals with MSC: 41A15, 65D07


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1982-0637290-3
Article copyright: © Copyright 1982 American Mathematical Society

American Mathematical Society