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

DOI:
https://doi.org/10.1090/S0025-5718-1978-0481754-1

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 one-sided 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.

**[1]**D. L. BARROW, C. K. CHUI, P. W. SMITH & J. D. WARD, "Unicity of best 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-splines. A survey," in*B**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 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 -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 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 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)**

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DOI:
https://doi.org/10.1090/S0025-5718-1978-0481754-1

Article copyright:
© Copyright 1978
American Mathematical Society