Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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



Unicity of best mean approximation by second order splines with variable knots

Authors: D. L. Barrow, C. K. Chui, P. W. Smith and J. D. Ward
Journal: Math. Comp. 32 (1978), 1131-1143
MSC: Primary 41A15
MathSciNet review: 0481754
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ S_N^2$ denote the nonlinear manifold of second order splines defined on [0, 1] having at most $ N$ interior knots, counting multiplicities. We consider the ques tion of unicity of best approximations to a function $ f$ by elements of $ S_N^2$. Approximation relative to the $ {L_2}[0,1]$ norm is treated first, with the results then extended to the best $ {L_1}$ and best one-sided $ {L_1}$ approximation problems. The conclusions in each case are essentially the same, and can be summarized as follows: a sufficiently smooth function $ f$ satisfying $ f'' > 0$ has a unique best approximant from $ S_N^2$ provided either $ \log f''$ is concave, or $ N$ is sufficiently large, $ N \geqslant {N_0}(f)$; for any $ N$, there is a smooth function $ f$, with $ f'' > 0$, having at least two best approximants. A principal tool in the analysis is the finite dimensional topological degree of a mapping.

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

  • [1] D. L. BARROW, C. K. CHUI, P. W. SMITH & J. D. WARD, "Unicity of best $ {L_2}$ approximation by second-order splines with variable knots," Bull. Amer. Math. Soc., v. 83, 1977, pp. 1049-1050. MR 0447889 (56:6199)
  • [2] C. DE BOOR, "Splines as linear combinations of B-splines. A survey," in Approximation Theory. II (G. G. Lorentz, C. K. Chui, L. L. Schumaker, Eds.), Academic Press, New York, 1976, pp. 1-47. MR 0467092 (57:6959)
  • [3] C. DE BOOR, "On the approximation by $ \gamma $ polynomials," in Approximation with Special Emphasis on Spline Functions (I. J. Schoenberg, Ed.), Academic Press, New York, 1969, pp. 157-183.
  • [4] D. BRAESS, "On the nonuniqueness of monosplines with least $ {L_2}$-norm,' J. Approximation Theory, v. 12, 1974, pp. 91-93. MR 0358149 (50:10614)
  • [5] C. K. CHUI, P. W. SMITH & J. D. WARD, "On the smoothness of best $ {L_2}$ approximants from nonlinear spline manifolds," Math. Comp., v. 37, 1977, pp. 17-23. MR 0422955 (54:10939)
  • [6] S. KARLIN, C. A. MICCHELLI, A. PINKUS & I. J. SCHOENBERG, Studies in Spline Functions and Approximation Theory, Academic Press, New York, 1976. MR 0393934 (52:14741)
  • [7] G. MEINARDUS, Approximation of Functions: Theory and Numerical Methods, Translated by L. L. Schumaker, Springer-Verlag, New York, 1967. MR 0217482 (36:571)
  • [8] J. T. SCHWARTZ, Nonlinear Functional Analysis, Courant Inst, of Math. Sci., New York, 1963. MR 0433481 (55:6457)
  • [9] P. W. SMITH, "On the smoothness of local best $ {L_p}$ spline approximations," in Approximation Theory, II (G. G. Lorentz, C. K. Chui, L. L. Schumaker, Eds.), Academic Press, New York, 1976, pp. 563-566. MR 0435675 (55:8633)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 41A15

Retrieve articles in all journals with MSC: 41A15

Additional Information

Article copyright: © Copyright 1978 American Mathematical Society

American Mathematical Society