Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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

 

 

On the problems of smoothing and near-interpolation


Author: Scott N. Kersey
Journal: Math. Comp. 72 (2003), 1873-1885
MSC (2000): Primary 41A05, 41A15, 41A29
Published electronically: May 1, 2003
MathSciNet review: 1986809
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In the first part of this paper we apply a saddle point theorem from convex analysis to show that various constrained minimization problems are equivalent to the problem of smoothing by spline functions. In particular, we show that near-interpolants are smoothing splines with weights that arise as Lagrange multipliers corresponding to the constraints in the problem of near-interpolation. In the second part of this paper we apply certain fixed point iterations to compute these weights. A similar iteration is applied to the computation of the smoothing parameter in the problem of smoothing.


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

  • 1. P. M. Anselone and P. J. Laurent, A general method for the construction of interpolating or smoothing spline-functions, Numer. Math. 12 (1968), 66–82. MR 0249904
  • 2. M. Atteia, Fonctions spline avec contraintes linéaires de type inégalite, Congrès de l'AFIRO, Nancy, Mai (1967).
  • 3. V. Barbu and Th. Precupanu, Convexity and optimization in Banach spaces, 2nd ed., Mathematics and its Applications (East European Series), vol. 10, D. Reidel Publishing Co., Dordrecht; Editura Academiei Republicii Socialiste România, Bucharest, 1986. MR 860772
  • 4. Carl de Boor, A practical guide to splines, Applied Mathematical Sciences, vol. 27, Springer-Verlag, New York-Berlin, 1978. MR 507062
  • 5. P. Copley and L. L. Schumaker, On 𝑝𝐿𝑔-splines, J. Approx. Theory 23 (1978), no. 1, 1–28. MR 0510721
  • 6. Paul Dierckx, Curve and surface fitting with splines, Monographs on Numerical Analysis, The Clarendon Press, Oxford University Press, New York, 1993. Oxford Science Publications. MR 1218172
  • 7. J. W. Jerome and L. L. Schumaker, A note on obtaining natural spline functions by the abstract approach of Atteia and Laurent, SIAM J. Numer. Anal. 5 (1968), 657–663. MR 0252912
  • 8. Scott Kersey, Best near-interpolation by curves: existence, SIAM J. Numer. Anal. 38 (2000), no. 5, 1666–1675 (electronic). MR 1813250, 10.1137/S0036142999355696
  • 9. S. Kersey, Near-interpolation, Numerische Mathematik (2003) to appear.
  • 10. P.-J. Laurent, Construction of spline functions in a convex set, Approximations with Special Emphasis on Spline Functions (Proc. Sympos. Univ. of Wisconsin, Madison, Wis., 1969) Academic Press, New York, 1969, pp. 415–446. MR 0252932
  • 11. O. L. Mangasarian and L. L. Schumaker, Splines via optimal control, Approximations with Special Emphasis on Spline Functions (Proc. Sympos. Univ. of Wisconsin, Madison, Wis., 1969) Academic Press, New York, 1969, pp. 119–156. MR 0259435
  • 12. Christian H. Reinsch, Smoothing by spline functions. I, II, Numer. Math. 10 (1967), 177–183; ibid. 16 (1970/71), 451–454. MR 0295532
  • 13. Christian H. Reinsch, Smoothing by spline functions. I, II, Numer. Math. 10 (1967), 177–183; ibid. 16 (1970/71), 451–454. MR 0295532
  • 14. I. J. Schoenberg, Spline functions and the problem of graduation, Proc. Nat. Acad. Sci. U.S.A. 52 (1964), 947–950. MR 0167768
  • 15. Eugene V. Shikin and Alexander I. Plis, Handbook on splines for the user, CRC Press, Boca Raton, FL, 1995. With 1 IBM-PC floppy disk (5.25 inch; HD). MR 1470222
  • 16. Grace Wahba, Spline models for observational data, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 59, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1990. MR 1045442

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 41A05, 41A15, 41A29

Retrieve articles in all journals with MSC (2000): 41A05, 41A15, 41A29


Additional Information

Scott N. Kersey
Affiliation: Department of Mathematics, Case Western Reserve University, 10900 Eulcid Avenue, Cleveland, Ohio 44106-7085
Email: snk@po.cwru.edu

DOI: https://doi.org/10.1090/S0025-5718-03-01523-0
Keywords: Near-interpolation, smoothing splines, approximation
Received by editor(s): July 20, 1999
Received by editor(s) in revised form: September 21, 2001
Published electronically: May 1, 2003
Article copyright: © Copyright 2003 American Mathematical Society