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On the problems of smoothing and near-interpolation
Author(s):
Scott
N.
Kersey.
Journal:
Math. Comp.
72
(2003),
1873-1885.
MSC (2000):
Primary 41A05, 41A15, 41A29
Posted:
May 1, 2003
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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:
-
- 1.
- P. M. Anselone and P. J. Laurent, A general method for the construction of interpolating or smoothing spline-functions, Numerische Mathematik 12 (1968), 66-82. MR 40:3145
- 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, 3rd ed., Reidel, Dordrecht (1986). MR 87k:49045
- 4.
- C. de Boor, A Practical Guide to Splines, Springer Verlag, New York (1978). MR 80a:65027
- 5.
- P. Copley and L. L. Schumaker, On
 -splines, J. Approx. Theory 23 (1978), 1-28. MR 58:23258 - 6.
- P. Dierckx, Curve and surface fitting with splines, The Clarendon Press, Oxford University Press, New York (1993). MR 94m:65028
- 7.
- J. 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 40:6127
- 8.
- S. Kersey, Best near-interpolation by curves: existence, SIAM J. Numer. Anal. 38 (2000), 1666-1675. MR 2001m:41034
- 9.
- S. Kersey, Near-interpolation, Numerische Mathematik (2003) to appear.
- 10.
- P. J. Laurent, Construction of spline functions in a convex set, Approximation with special emphasis on spline functions, I. J. Schoenberg, ed., Academic Press, New York (1969), 415-446. MR 40:6147
- 11.
- O. L. Mangasarian and L. L. Schumaker, Splines via optimal control, Approximation with special emphasis on spline functions, I. J. Schoenberg, ed., Academic Press, New York (1969), 119-155. MR 41:4073
- 12.
- C. H. Reinsch, Smoothing by spline functions, Numerische Mathematik 10 (1967), 177-183. MR 45:4598
- 13.
- C. H. Reinsch, Smoothing by spline functions II, Numerische Mathematik 16 (1971), 451-454. MR 45:4598
- 14.
- I. J. Schoenberg, Spline functions and the problem of graduation, Proc. Nat. Acad. Sci. USA 52 (1964), 947-950. MR 29:5040
- 15.
- E. V. Shikin and A. I. Plis, Handbook on splines for the user, CRC Press, Boca Raton, FL (1995). MR 98f:65015
- 16.
- G. Wahba, Spline Models for Observational Data, CBMS-NSF Regional Conference Series in Applied Mathematics 59, SIAM, Philadelphia, PA, (1990). MR 91g:62028
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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:
10.1090/S0025-5718-03-01523-0
PII:
S 0025-5718(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.
Posted:
May 1, 2003
Copyright of article:
Copyright
2003,
American Mathematical Society
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