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), 11311143
MSC:
Primary 41A15
MathSciNet review:
0481754
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Abstract: Let denote the nonlinear manifold of second order splines defined on [0, 1] having at most interior knots, counting multiplicities. We consider the ques tion of unicity of best approximations to a function by elements of . Approximation relative to the norm is treated first, with the results then extended to the best and best onesided approximation problems. The conclusions in each case are essentially the same, and can be summarized as follows: a sufficiently smooth function satisfying has a unique best approximant from provided either is concave, or is sufficiently large, ; for any , there is a smooth function , with , having at least two best approximants. A principal tool in the analysis is the finite dimensional topological degree of a mapping.
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 D. L. BARROW, C. K. CHUI, P. W. SMITH & J. D. WARD, "Unicity of best approximation by secondorder splines with variable knots," Bull. Amer. Math. Soc., v. 83, 1977, pp. 10491050. MR 0447889 (56:6199)
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 C. DE BOOR, "On the approximation by polynomials," in Approximation with Special Emphasis on Spline Functions (I. J. Schoenberg, Ed.), Academic Press, New York, 1969, pp. 157183.
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 D. BRAESS, "On the nonuniqueness of monosplines with least norm,' J. Approximation Theory, v. 12, 1974, pp. 9193. MR 0358149 (50:10614)
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 C. K. CHUI, P. W. SMITH & J. D. WARD, "On the smoothness of best approximants from nonlinear spline manifolds," Math. Comp., v. 37, 1977, pp. 1723. MR 0422955 (54:10939)
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 G. MEINARDUS, Approximation of Functions: Theory and Numerical Methods, Translated by L. L. Schumaker, SpringerVerlag, New York, 1967. MR 0217482 (36:571)
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 P. W. SMITH, "On the smoothness of local best spline approximations," in Approximation Theory, II (G. G. Lorentz, C. K. Chui, L. L. Schumaker, Eds.), Academic Press, New York, 1976, pp. 563566. MR 0435675 (55:8633)
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DOI:
http://dx.doi.org/10.1090/S00255718197804817541
PII:
S 00255718(1978)04817541
Article copyright:
© Copyright 1978
American Mathematical Society
