B-splines and optimal stability

Peña, J.

doi:10.1090/S0025-5718-97-00897-1

B-splines and optimal stability
HTML articles powered by AMS MathViewer

by J. M. Peña PDF

Math. Comp. 66 (1997), 1555-1560 Request permission

Abstract:

It is proved that, among all nonnegative bases of its space, the B-spline basis is optimally stable for evaluating spline functions.

References

Carl de Boor, The exact condition of the $B$-spline basis may be hard to determine, J. Approx. Theory 60 (1990), no. 3, 344–359. MR 1042654, DOI 10.1016/0021-9045(90)90064-W
J. M. Carnicer and J. M. Peña, Shape preserving representations and optimality of the Bernstein basis, Adv. Comput. Math. 1 (1993), no. 2, 173–196. MR 1230256, DOI 10.1007/BF02071384
J. M. Carnicer and J. M. Peña, Least supported bases and local linear independence, Numer. Math. 67 (1994), no. 3, 289–301. MR 1269498, DOI 10.1007/s002110050029
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), no. 6, 633–654. MR 1305911, DOI 10.1016/0167-8396(94)90056-6
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.
R. T. Farouki and T. N. T. Goodman, On the optimal stability of the Bernstein basis, Math. Comp. 65 (1996), no. 216, 1553–1566. MR 1351201, DOI 10.1090/S0025-5718-96-00759-4
R. T. Farouki and V. T. Rajan, On the numerical condition of polynomials in Bernstein form, Comput. Aided Geom. Design 4 (1987), no. 3, 191–216. MR 917780, DOI 10.1016/0167-8396(87)90012-4
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.
Larry L. Schumaker, Spline functions: basic theory, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1981. MR 606200