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B-splines and optimal stability
Author(s):
J.
M.
Peña.
Journal:
Math. Comp.
66
(1997),
1555-1560.
MSC (1991):
Primary 65D07, 41A15
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Abstract:
It is proved that, among all nonnegative bases of its space, the B-spline basis is optimally stable for evaluating spline functions.
References:
- [1]
- C. de Boor, The exact condition of the B-spline basis may be hard to determine, J. Approx. Theory 60 (1990), 334-359. MR 91h:65023
- [2]
- J. M. Carnicer and J. M. Peña, Shape preserving representations and optimality of the Bernstein basis, Adv. Comput. Math. 1 (1993), 173-196.MR 94i:65138
- [3]
- J. M. Carnicer and J. M. Peña, Least supported bases and local linear independence, Numer. Math. 67 (1994), 289-301. MR 95d:65012
- [4]
- J. M. Carnicer and J. M. Peña, Totally positive bases for shape preserving curve design and optimality of B-splines, Comput. Aided Geom. Design 11 (1994), 633-654. MR 95i:65033
- [5]
- J. M. Carnicer and J. M. Peña, Total positivity and optimal bases, Total positivity and its applications (M. Gasca, C.A. Micchelli, eds.), Kluwer Academic Press, Dordrecht, 1996, pp. 133-155. CMP 97:04
- [6]
- R. T. Farouki and T. N. T. Goodman, On the optimal stability of the Bernstein basis, Math. Comp. 65 (1996), 1553-1566. MR 97a:65021
- [7]
- R. T. Farouki and V. T. Rajan, On the numerical condition of polynomials in Bernstein form, Comput. Aided Geom. Design 4 (1987), 191-216. MR 89a:65028
- [8]
- T. Lyche, Condition Numbers for B-splines, Numerical Analysis 1989 (D. F. Griffiths and
G. A. Watson, eds.), Longman Scientific and Technical, Essex, 1990, pp. 182-192. CMP 91:17 - [9]
- L. L. Schumaker, Spline Functions: Basic Theory, John Wiley and Sons, New York, 1981.MR 82j:41001
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Additional Information:
J.
M.
Peña
Affiliation:
Departamento de Matem{á}tica Aplicada, Universidad de Zaragoza, 50009 Zaragoza, Spain
Email:
jmpena@posta.unizar.es
DOI:
10.1090/S0025-5718-97-00897-1
PII:
S 0025-5718(97)00897-1
Keywords:
B-splines; optimal stability,
condition number,
nonnegative matrices,
partial ordering
Received by editor(s):
May 10, 1995
Received by editor(s) in revised form:
July 29, 1996
Additional Notes:
This work was partially supported by the Spanish Research Grant DGICYT PB93-0310 and by the EU project CHRX-CT94-0522.
Copyright of article:
Copyright
1997,
American Mathematical Society
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