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

Barrow, D. L.; Chui, C. K.; Smith, P. W.; Ward, J. D.

doi:10.1090/S0025-5718-1978-0481754-1

Unicity of best mean approximation by second order splines with variable knots
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by D. L. Barrow, C. K. Chui, P. W. Smith and J. D. Ward PDF

Math. Comp. 32 (1978), 1131-1143 Request permission

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

D. L. Barrow, C. K. Chui, P. W. Smith, and J. D. Ward, Unicity of best $L_{2}$ approximation by second-order splines with variable knots, Bull. Amer. Math. Soc. 83 (1977), no. 5, 1049–1050. MR 447889, DOI 10.1090/S0002-9904-1977-14377-2
Carl de Boor, Splines as linear combinations of $B$-splines. A survey, Approximation theory, II (Proc. Internat. Sympos., Univ. Texas, Austin, Tex., 1976) Academic Press, New York, 1976, pp. 1–47. MR 0467092

Approximation with Special Emphasis on Spline Functions

Dietrich Braess, On the nonuniqueness of monosplines with least $L_{2}$-norm, J. Approximation Theory 12 (1974), 91–93. MR 358149, DOI 10.1016/0021-9045(74)90061-6
Charles K. Chui, Philip W. Smith, and Joseph D. Ward, On the smoothness of best $L_{2}$ approximants from nonlinear spline manifolds, Math. Comp. 31 (1977), no. 137, 17–23. MR 422955, DOI 10.1090/S0025-5718-1977-0422955-7
Samuel Karlin, Charles A. Micchelli, Allan Pinkus, and I. J. Schoenberg (eds.), Studies in spline functions and approximation theory, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1976. MR 0393934
Günter Meinardus, Approximation of functions: Theory and numerical methods, Expanded translation of the German edition, Springer Tracts in Natural Philosophy, Vol. 13, Springer-Verlag New York, Inc., New York, 1967. Translated by Larry L. Schumaker. MR 0217482
J. T. Schwartz, Nonlinear functional analysis, Notes on Mathematics and its Applications, Gordon and Breach Science Publishers, New York-London-Paris, 1969. Notes by H. Fattorini, R. Nirenberg and H. Porta, with an additional chapter by Hermann Karcher. MR 0433481
Philip W. Smith, On the smoothness of local best $L_{p}$ spline approximations, Approximation theory, II (Proc. Internat. Sympos., Univ. Texas, Austin, Tex., 1976) Academic Press, New York, 1976, pp. 563–566. MR 0435675